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By A New METHOD, FOUNDED ON THE TRUE SYSTEM OF 
str IsAAc NEWTON, WITHOUT THE USE OF 
INFINITESIMALS OR LIMITs. 


By CG. P. BUCKINGHAM, 


AUTHOR OF THE PRINCIPLES OF ARITHMETIC} FORMERLY ASSISTANT PROFESSOR OF 
NATURAL PHILOSOPHY IN THE U.S. MILITARY ACADEMY, AND PROFESSOR OF 
MATHEMATICS AND NATURAL PHILOSOPHY IN KENYON COLLEGE, OHIO, 


CHICAGO: “JLr 
SRC RIGGS & COMPANY 
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“The student of mathematics, on passing from the lower 
branches of the science to the infinitesimal analysis, finds 
himself in a strange and almost wholly foreign department 
of thought. _He has not risen, by easy and gradual steps, 
from a lower to a higher, purer, and more beautiful region of 
scientific truth. On the contrary, he is painfully impressed 
with the conviction, that the continuity of the science has 
been broken, and its unity destroyed, by the influx of prin- 
ciples which are as unintelligible as they are novel. He 
finds himself surrounded by enigmas and obscurities, which 
only serve to perplex his understanding and darken his aspi- 
rations after knowledge.” * 

He finds himself required to ignore the principles and 
axioms that have hitherto guided his studies and sustained 
his convictions, and to receive in their stead a set of notions 
that are utterly repugnant to all his preconceived ideas of 
truth. When he is told that one quantity may be added to 
another without increasing it, or subtracted from another 
without diminishing it — that one quantity may be infinitely 


small, and another infinitely smaller, and another infinitely 


* Bledsoe — Philosophy of Mathematics, 


lV PREFACE. 


smaller still, and so on ad ¢njinitum — that a quantity may 
be so small that it cannot be divided, and yet may contain 
another an indefinite, and even an zzfimite, number of times — 
that zero is not always nothing, but may not only be some- 
thing or nothing as occasion may require, but may be doh at 
the same time in the same equation— that two curves may 
intersect each other and yet both be tangent to a third curve 
at their point of intersection,* it is not surprising that he 
should become bewildered and disheartened. Nevertheless, 
if he study the text books that are considered orthodox in 
this country and in Europe he will find some of these 
notions set forth in them all; not, indeed, in their naked 
deformity, as they are here stated, but softened and made as 
palatable as possible by associating them with, or concealing 
them beneath, propositions that are undoubtedly true. 

It is, indeed, strange that a science so exact in its results, 
should have its principles interwoven with so much that is 
false and absurd in theory; especially as all these absurdi- 
ties have been so often exposed, and charged against the 
claims of the calculus as atrue science. It can be accounted: 
for only by the influence of the great names that first adopted 
them, and the indisposition of mathematicians to depart 
from the simple ideas of the ancients in reference to the 
attributes of quantity. They regard it as merely inert, 
elther fixed in value or subject only to such changes as may 
be arbitrarily imposed upon it. But when they attempt to 
carry this conception into the operations of the calculus, 


and to account for the results by some theory consistent with 


* As in the case of exvelopes, 


PREFACE, Vv 


this idea of quantity they are inevitably entangled in some 
of the absurd notions that have been mentioned. Many 
efforts have indeed been made to escape such glaring incon- 
sistencies, but they have only resulted in a partial success 
in concealing them. 

To clear the way for a logical and rational consideration 
of the subject, we must begin with the fundamental idea of 
the conditions under which quantity may exist. We must, 
for the purposes of the calculus, consider it not only as ca- 
pable of being increased or diminished; but also as being 
actually in a state of change. It must (so to speak) be z7¢al- 
zze@, so that it shall be endowed with éendencies to change its 
value; and the’ rate and direction of these tendencies will 
be found to constitute the ground work of the whole system. 
The differential calculus is the SCIENCE OF RATES, and its 
peculiar subject is QUANTITY IN A STATE OF CHANGE. 

It is an error, therefore, to suppose, as has often been 
said, that the “veductto ad absurdum,” or “ method of ex- 
haustion,” of the ancient mathematicians, contained the 
germ of the differential calculus. This hallucination has 
arisen from the same source as the false notions before 
mentioned. That peculiar attribute of quantity upon which 
the transcendental analysis was built, never found a place 
among the ideas of the Greek Philosophers; and even Leib- 
nitz, the competitor of Newton for the honor of the inven- 
tion, and who was the first to construct a system of rules 
for the analytical machinery of the science, never got beyond 
the ancient conception of the conditions of quantity, and, 


therefore, gave, says Comté, “an explanation entirely erro- 


vi PREFACE. 


neous ”— he never comprehended the true philosophical 
basis of his own system. 

The only original birth-place of the fundamental idea of 
quantity which forms the true germ of the calculus, was in 
the mind of the immortal Newton. Starting with this idea, 
the results of the calculus follow logically and directly 
through the beaten track of mathematical thought, with that 
clearness of evidence which has ever been the boast of 
mathematics, and which leaves neither doubt nor distrust 
in the mind of the student. 

To develop this idea is the object of this work. 


C. P. BUCKINGHAM 


CHICAGO, 0/271.-1,0LO7 51 


CoNnTENTS. 


PE NTIEROED UG ELON: 

PAGE, 

Objects of Mathematical Study among the Ancient Philosophers... 11 
WEGENER INC ERGOT LCS Se ra se oe EEE 32, 2 eek a ok st cle 12 
Pee ieTeN aA AICIINS «om ho roe eit Sg ak ce te eee betel ec 14 
wo wrenetal wlethodes.3 f22tee* 2S TO he SE Cd een, Lie ie ed 16 
Boreal Merbndy 2S 2 aa cose eee ee Sue Ee oe 18 
Resiicarertrie while the Method 1s-Falsé: 3. sae = Soa es eke 20 
Debord Oe Laniice & se oat os ten ye eee ns OG GRE eee ee eee o 22 
Newtow 5 Defense Of his Leminia tt ese Fae ee 23 
Opinions of Comtés Lbagratige and, Berkeley... + 5. £io lol ces 24 
Dusveaments error or both oystems. — 20. - 2. ona sous shes ces 31 
Lier prueivrernodion. Newton: cater oon. cece lew ea aa oe ceoe 33 
Die MOUnGALOI Of tual: Method =~ . 2c setee se. Son See SLL ae esses 34 
Explanatory Letter of the Author---__- I eke es Oe mk fe: 

PA Riis ok: 
DIFFERENTIAL CALCULUS. 

SuCtions ls. qLetnitions and birst. Principles 2 -__o Son aecann oe se 49 
Warinnieamenneds 2eo.c- ent). en se Jo es eee 49 

Prater Gtey ALICMON tries meso. Veter ye © Aira kc oe ota rohe S 50 
Peretti ets ye en enn pe a re vem ee rae Oo ree 51 

RO RtLiSe ee oe rs eae. Oe Sal oe ee Pee hs, feet Se 52 

Pa IIee Sey Sees ker ere ee Oe os 53 

mec ries uietCOration. Of MUNCHONS 4. senso. ee on 2d 5 3 56 
SI eAeCUCREMCCT Attia shoe se tetas ect Ft 56 
Differential of a Function Consisting of Terms ------------ 58 
MrewiscOriceoraiGg Pernice kuere mad. Be. ces 60 
Differential. of a Product of Two Variables.---....-------- 63 
meriet Cala tie rAlOn Sw ee ee. fe ycte son At elo 65 


jirerentiais of ractions «i024 02-2 Pn ho Eee eh ee lard 68 


vill . CONTENTS. 


PAGE. 
Differential of the Power of*a Variabley.o. vane e ees eee 71 
# + sRoot me Mh. Peo, eae see 73 
‘ “Function of Another Function ......:-. 74 

SECTION III. “Suecessive Differentials $22 eee ee Fae ras A 
Maclaurin’s Theorem..-::<.- ---.¢slccn eaeee eee eee 85 
Taylor's @ heorem=s)0o 8 22 ee er :. See 89 
Identity of Principle in Both Theorems (note)_------------ 93 
SECTION [V., Maxima and Minima .---Cs22. 2522.22. 95 
Methodzof Finding by Substitution.22. 2322. 2>-2_ eee 96 
Meahing<of ra, Maximum S22 = oak eee eee ahd 97 
Method of Finding by the Second Differential_..-._------- 98 
Use of: Other Difterential.Coetitients=_~ 2252.0) eee 99 
Examples Ilustrating Different Cases.-.. -. <-eeacee ee eee 100 
Analytical Demonstration of the General Rule. ......_..... 105 
SECTION V. Application of the Calculus to the Theory of Curves. 125 
Definition of a:Lines =. a ee ek ee 125 
Why the-Line-Becomes’a Cutye 2.22 ee eee 126 
Direction of the Danpent Line 222-5 ya ree 127 
Real Meaning of the Differential Equation -_---.-2=-2.-2- 129 
Sign of ‘the First Differential Coéficiente)  ee-= sees eee 132 
SecTION VI. Differentials of Transcendental Functions. ..--.-- Pie te 
Differential.of ‘eos 2 node kee ee ee eee 135 
“f of the Logarithm of @ Vanables. oss esses 137 
a “"~ <oiné ofian Aten s23, 2. sss eee 139 
ss or Cosine ofan Arce ss.. 5. 140 
a « Tangent ofan: Are 22 ee eee 141 
Sy “ . Secant of, an Are-22>-222232 eee 142 
LN “__ -Versed Sine. ofan Arcs eee eee 143 
A fe ATO 34 2252S Shs ee ee 143 
Signification of the Differentials of Circular Functions __-__- 144 
Values of Trigonometrical Liness2s- eee sa eee eee 148 
SECTION VII. Tangent and Normal Lines to Algebraic Curves.__ 150 
Length of ‘Subtangent to any Curve.... 3-2. o ree 152 
4 ‘« Tangent ‘> OP ite dno 153 
a wks ou TOs Ines = SU Reet hes oe ee 163 
“Normal A Ff oo nie oS Line ae ee 154 
AppHeation of the formulas 22.5 oco0 c= oe ae 154-156 
SECTION VIII. . Differentials of Curves’. 2222-52 - oe oe 157 
Differential Plane Surfaces Bounded by Curves._-_.-------- 158 


«6 Surfaces. of Revolutioniz->.cs.ee. eee ee 162 


CONTENTS. 1X 


: PAGE, 

Witterential Solids of. Revulotion 2 }<2 02 osc e cs ont pee =~ TOs 

DUC LION ¢Lacuwe t Olats GUiEVeSUcee. oc re oe Dee eens ot See ae 168 
mangzents:to Polar-Curmes= i. Sass tet soe. «o~ =~ - == 168 
Differential’ of ‘the Arc of/a Polar. Gurvers2. -—<.28_ J. =. L7I 
Puibtanpent tie: P Ole Curve 22 oe a eae oe naa wet rye 
Tangent : * CR ee nee wt renee r7r 
Subnormal ‘“ es De Se Cae hee ek yas ee aa 172 
Normal . $ Ee a AP ae ne RS 172 
Surrice: bounded by.a- Polar Curve 2222 2-2 oe 2 see 173 

SY ATCT ia So ASE A Raced NAA BEI Sepa RO eee Te a 173 
SPINOR ORURCUCS cin rea ns ain ae eee disciga's ah 174 
Peper tie pin tns.~ 2 tata cea ee ens orn i a ee 176 
Omri tia ae as ae ee a eee dees a ae <2 es SAS 179 
MeOTION de ~ Asymntotes roi o at See a ee 182 
Laka g Ko We he Ye Wie Canc hk ape ah et te aie aie, SAM Ae 5, Se 183 
TORRID IOS ese ts erg crete pets kA eel Sa A eer 183-186 
SECTION XI. Signification of the Second Differential-._.-..----- 187 
Siegen of the Second Differential Coefficient £-- 4.2 (Ss. <.1--- 188 
~Value ‘“ . ~ Re he ey Ee ee, = a 190 
DERTION: Sti eervatnre OL Mantes es esate ee ot tL oo als cous 192 
Dboeurercn Curvature arenes eas een hee Se ee 192 
(APRAOCOS KOT VES Speke ewe Sean keys Pee fas 193 
Comerants.in the. Hquationof a:Curve S....-.--2--2-.t--<- 195 

WSC retGlee LC) a) ENG ce ae ee Sete ren Oe 199 
Isai ieGHitna ite eee pee toe eee 3 ks oets con 202 

SEC TEON  & Lic eth VOL Case eine a oer oe oa 207 
Pleperiies Of ties Myo iter aa wen oe ee een tS 207 
PorPme-the Equation sof: Mie Wyoute co. 5. 20.202. fect e 210 

De TIONS Ci aenVelopes. - S ere e t o e  e wS 216 
Peseta Ol, ean VOLO Pe: Lomeenad (ooo oe ea ek 218 

Bre wato a italieis Maton --bneees oo ee Fa oe las 216 
SECTION XV. Application of the Calculus to the Discussion of Curves 229 
fie Cyeloids sea gs s ee te ee ee aie cae dy sls uk 229 

Ce periieer On tue VelOIG 22 06 — soa a= doin nce oe Stee 230 
Domai iiiae Cine ie "Le Sees o ek sue 238 

Pe One ey Leen Siloigi LOMG2e5 205 toe sacs 2k el 558. 243 
Rigedinia CaM Wit tebe yoo a oe eo we oe See 243-248 

Saha) Omen Te eae RA Feros, SER 2 Bo ae ok a oe eS 248 
Mie Ot Sena. oo os See « den = Uy Seta} 253 

Di ites Gin ts eres fer ed Se Serge ee es wee 254 


xe CONTENTS. 


PARs wes 


INTEGRAL CALCULUS. 


PAGE 

SEETION J. . “Principles -of "Integration <. 22 oo eee ee 261 

Integration of Compound Differential Functions....--.---- 263 

FSS Monomial a SS es Po ee ee 263 

3 Particular Binomial Differentials --.-....--- 266 

xs Rational Fractious! se 52.8 wee koe eee 207 

sd Between, limits =~ ei eee oo ee 277 

is by Sertes Stee pee ee Se ee ee 278 

of Differentials-ofCircnlan Arcelor 279 

SECTION II. Integration of Binomial Differentials .......------_- 282 

Integration’ of Particular omnes sss) — eee. eee 283-288 

z by..Parts sare ari ered aes See ee 291 

Formulasdor Reducing Eixponeniss 22. eee 293-303 
SECTION III. Application of the Calculus to the Measurement of 

Geometrical “Mapnitudesc. 22) see. eee eee 304 

Recwheation of Ciurves:. oS oe eo a ee ae 305 

Ouddtature of (Curves espe ao. Pe ee ee 212 

Suraces of: Revolution.c<.. <-eoeoe eaea> cee een eee ee 325 

Cubature of Solids 22 see ok cae eae an ee Soe 329 


APP a NgLD Lek, 


GEOMETRICAL FLUXIONS. 


Principles of the Calculus Applied Directly to Geometrical Magnitudes 337 


To: Find the Area of a) Circle. 3. 22 sot eee - ee 338 
So US (Convex Surlace Otta, CONG se sae a. era ene 339 
“> se S* NMolume ofa Cone 2 cates oo eee eee 340 
tern <® -Avea of the Surface’ofca Sphere-<- ae ee 341 


“ce 66 6c 


Volume of a Spheteseis si sccec ee -te wee eee eee 342 


INTRODUCTION. 


THE PHILOSOPHY OF THE CALCULUS. 


Among the ancient philosophers, the objects of mathemat- 
ical study were confined exclusively to the solution of deter- 
minate problems — that is, every quantity concerned had a 
determinate value, either known or unknown. Devoting 
themselves principally to Geometry, they sought to deter- 
mine the exact measurement of lines, surfaces, solids and 
angles, in terms of fixed and known quantities. 

The later methods of the algebraists did not change the 
ultimate object of their researches. All their problems 
were still determinate. Their conditions were definite, and 
the result certain. This, which may properly be character- 
ized as the s/at#c phase of mathematics, continued for two 
thousand years to guide, control and circumscribe the labors 
of the mathematical student. There was but little advance 
in the discovery of mathematical truth; none had the bold- 
ness to strike out a new method of investigation, or apply 
themselves to the solution of any but determinate problems. 
Algebra had indeed been successfully applied to geometry, 
but it was only the analytical method of stating arguments 
that had been used in ordinary language for centuries. 


X1l INTRODUCTION. 


Such was the condition of the science up to the time 
when the brilliant genius of Descartes seized upon a new 
idea, and boldly followed its lead until he developed a sys- 
tem whose results have astonished and delighted the world. 

Breaking away from the idea of determinate values and 
. absolute conditions, he adopted that of dependent condi- 
tions and relative values, which no longer fixed unchangea- 
bly the quantities sought, but gave them a wide range, so 
that within certain limits they could have all possible values. 
Hence they were called varzables, while those quantities 
whose values were fixed were called constants. 

In every equation containing a single unknown quantity 
the value of that quantity is absolutely fixed by the condi- 
tions expressed in the equation. If we have two unknown 
quantities, and two equations, or sets of conditions, both 
values are still fixed. If the higher power of the unknown 
quantity is involved, the number of values is greater, but 
they are equally fixed and certain. This idea of fixedness 
of value underlies all algebraic operations of an ordinary 
kind. 

Now suppose we have “wo unknown quantities in ove equa- 
tion, with no other conditions given than those expressed by 
the equation itself. In that case the values of both quanti- 
ties are absolutely indeterminate. But if we know or 
assume any specific value for one we can at once determine 
the corresponding value of the other; so that while the 
equation will give the independent value of neither quan- 
tity, it will give the szmultaneous values of both ; and these 
values will have a certain range or locus, which is in fact the 
true solution of the equation — the path, so to speak, through — 
which the simultaneous values range. 

In some equations the range of values‘is limited for both 
variables, so that if a value be assigned to one beyond the 
limit, that of the other becomes imaginary; in other cases 


INTRODUCTION. Xiil 


the value of one only is limited, while in others again the 
values of both are absolutely unlimited; any value of one 
giving a corresponding real value for the other. 

Since the values of these variables are thus dependent on 
each other, the equation expressing this dependence may be 
considered as containing the /azw of their mutual relations, 
and the fundamental ideas of Descartes was to exhibit in his 
equation the conditions or law which confined the two vari- 
ables to their prescribed range of values. This idea was 
something new, distinct and well defined, and a clear:ad-_ 
vance beyond the methods of the ancients. 

But the labors of Descartes would have been of little 
value had he proceeded no farther than we have indicated. 
In fact this was but a part of his invention, of which the 
specific object was a method of investigating questions 
of Geometry. To complete this purpose, he devised a new 
and beautiful method of representing magnitudes, to which 
his algebraic equations could be applied. In algebra, all 
_values are estimated by their remoteness from zero. In 
order to make a general application of algebraic symbols to 
geometry, it was necessary that the value of every line rep- 
resented by his variables should be estimated from a com- 
mon origin corresponding with zero; and as every point in 
a plane surface requires two values to fix its position, two 
such origins became necessary to his system, in order to 
represent plain figures; and these were found in two right 
lines, lying in the plane of the object to be represented, and 
intersecting in a known angle — generally a right angle. 
From these two lines all values or distances to points were 
estimated ; the positive on one side and the negative on the 
other of each line; while for points zz the lines, one of the 
values would of course be zero. 

Having then a method of representing the position of a 
point by algebraic symbols, it was easy to apply his analysis 


X1V INTRODUCTION. 


to the representation of lines, by making the locus or range 
of simultaneous values of the variables to correspond with 
the locus of the points in the line —that is, with the line 
itself 

Thus the method of Descartes was two-fold — the alge- 
braic idea of two variables in one equation with a range of 
simultaneous vaiues, and the geometrical idea of coordinate 
representation, and these two being adapted to each other, 
united to form a method of investigating, in an easy and sim- 
ple manner, questions of geometry which had taxed the 
utmost powers of the ancients. 

Upon the foundation thus laid by Descartes has arisen 
the Differential Calculus. Not that the calculus in its purely 
abstract conception is especially related to geometry. On 
the contrary its analysis is adapted to investigations in all 
those questions where the quantities are variable and the 
conditions can be analytically expressed. But it was in con- 
nection with problems of geometry that its methods were first 
discovered, and fora long time applied through the Cartesian 
system; and even now geometry is the principal arena upon 
which the triumphs of the calculus are displayed. 

The invention itself has many peculiarities both in its 
history and substance. It was not a result produced by 
means of ordinary scientific investigation — by a discovery 
of fundamental principles and a careful elaboration of those 
principles until they grew into a perfect science. On the 
contrary these results appeared rather as remarkable phe- 
nomena, discovered more by accident than by- logical deduc- 
tion. Newton seems sie to have had an indistinct per- 
ception of the principles lying at the foundation, but he has 
nowhere given aclear and satisfactory account of them; while 
the explanation given by Leibnitz proved his utter ignorance 
of the true theory of his own system. 

Thus was a most important branch of mathematics invented 


INTRODUCTION. XV 


a Bf 
=>. 


almost simultaneously by two of the most distinguished@rven 
of the age, without any clear and fundamental principle for 
its foundation. Its operations were accepted as undoubtedly 
reliable, zof because its principles were sound, but because 
ats results were undeniably true, ‘Vhis could not be disputed, 
and hence mathematicians were not so eager, at first, to 
establish a logical basis for the new science, as to extend its 
Operations into new fields of discovery. Attempts were, at 
length, made to assign a rational principle which would 
account for these extraordinary results, but although each 
theory has had for its advocates many of the most distin- 
guished mathematicians, yet each one has had as many 
and as distinguished opponents. No one has secured the 
universal approval of the scientific world, and, therefore 
no cne was founded in mathematical truth; for no pro- 
position is worthy of the name that does not command 
the unqualified assent cf every mind by which it is fully 
comprehended. | 

The conceptions of the calculus were so subtle, its pro- 
cesses so mysterious, and its results so astounding that 
mathematicians began to lock upon its ideas as not subject 
to the ordinary laws of thought and the rigid rules of 
logic. The inconsistencies and absurdities which were 
often propounded were regarded as only mysterious and 
zncomprehensible ; when quantities refused to obey the laws 
that had always hitherto controlled them, they were called 
infinitesimals, and thus released from all subjection to 
establish axioms. Of course theories were not wanting, but 
they did little more than give variety to what was, after all, 
the same compound of false premises and illogical conclu- 
sions. We are told by M. Comte that the science is as yet 
in a “ provisional state,” and that it is necessary to study all 
the principal methods in order to have even an approximate 
understanding of it. 


Xvl INTRODUCTION. 


I shall, however, confine myself to the consideration of 
the two principal conceptions attributed to its inventors, 
and to some more recent modifications of them. 

_The advocates of these two methods have approached the 
subject from the same direction, but the theories involved in 
their demonstrations are fundamentally different. These I 
propose to examine, and to show that as theories they are 
fatally defective; that a fundamental error underlies every 
form in which they have been proposed, and muws¢ vitiate 
any theory based upon the leading idea through which they 
approach the subject. 

By the invention of Descartes very many geometrical 
problems were beautifully solved, but there were some for 
which equations between the direct functions of the varia- 
bles would not suffice; such as the length of a curve, the 
amount of its curvature, and others of a similar kind. To 
form equations in which such values. could be introduced, 
it became necessary to represent the variables zzdrectly, 
using, instead of the actual value, a /wmction of that value. 
This function is what is called the a@fferential of the variable, 
and the true philosophical relation which it sustains to its 
actual value has been the subject of controversy from the © 
beginning. The particular application of the calculus, which 
will most clearly and exactly illustrate the various systems, 
is Zo determine from the equation of the curve the direction of tts 
tangent at any point. This process involves the fundamental 
principles of the science, and an examination of it will 
afford the best means of investigating the different theories 
that have been advanced to account for the exact truth of 
the results obtained. 

Let APC (Fig. A) be a curve, and let AM and AN be the 
coordinate axes to which it is referred. Suppose the line SD 
to be drawn tangent to the curve at the point P. The 
problem is to find from the equation of the curve an ex- 


INTRODUCTION. 2 XVil 


pression for the value of the 
angle PSB. Now the tangent 
of this angle is 

PB 

‘SB 
S being the point of intersec- 
tion of the tangent line with the 
axis of abscissas, and B the 
foot of the ordinate through - Fig. A. 
the point of contact. Let BB’ be an increment added to the 
abscissa, and B’P’ the ordinate corresponding to the abscissa 
thus increased. Draw PE parallel to AM, intersecting B’P’ 
in E; then P’E is the corresponding increment of the ordi- 
nate BP. Draw the cord P’P, and let D be the point where 
the tangent intersects the ordinate B’P’. Since 

PY seit; 
' “SB” PE 

the problem is reduced to finding -the ratio of PE to DE. 
Now we can easily find from the equation of the curve, the 
line P’E corresponding to any increment BB’ or PE of the 
abscissa AB But DE is the line we need, and to pass from 
P’E to it is the specific part of the operation which involves 
the fundamental principles of the Calculus. The two prin- 
cipal methods of doing this we will now examine. 


THE INFINITESIMAL METHOD. 


The fundamental propositions or principles of this method 
te: First. “ That we may take indifferently the one for the 
other, two quantities which differ from each other by an infi- 
nitely small quantity, or (what is the same thing) that a 
quantity which is increased or diminished by another infinity 
less than itself can be considered as remaining the same. 
Second. That a curved line may be considered as an assem- 
2 


xvlil INTRODUCTION. 


blage of an infinity of right lines each infinitely small, or 
(what is the same thing) as a polygon with an infinite num- 
ber of sides infinitely small, which determine, by the angle 
which they make with each other, the curvature of the 
line.” | 
In order to understand clearly how these propositions 
apply to the solution of the problem we will consider it 
analytically. Suppose the equation of the curve to be 
yas? (1) 
If we take AB=x we have BP=y; and if we add to x an 
increment, BB’ (which we will call 2), the corresponding 
ordinate will be P’B’ , which we will designate by y’, and 
P’E will be equal to y’—y. Having added 2 to x, the new 
state of the Sek will be 
y =(x+h)? =x? +2hx+h? mes, 
Subtracting (1) ge (2) we have 
y —y=2hx+h? 
or dividing by 2 
ff é - la 
. 5 “ =2x+h as 
Now as # diminishes in value, this ratio approaches the 
value of 2x; and if Zis made infinitely small, it may by 
the first proposition above stated, be set aside as not affect- 
ing the value of the second member of the equation, which 
then becomes 2x. In the meantime the curve PP’ will have 
become infinitely small, and therefore by the second propo- 
sition may be considered a straight line, coincident with the 
tangent, that is with PD, so that P’E has become the same 
as DE, and the equation 


f 


PE 


=2x-+h 


has become 
DE 


PE —2X 


INTRODUCTION, X1X 


and thus the angle made by the tangent line with the axis 
of abscissas is determined. Such is the infinitesimal 
theory. 

The propositions upon which this theory is founded can- 
not be admitted as true, neither is the demonstration con- 
clusive. To whatever extent Z may be reduced, even to an 
infinitesimal (if it is possible to conceive such a thing), the 
two expressions 2x and 2x-+/ can never be equal so long as 
his anything. We must either admit this or abandon the use 
of our reason. If % becomes nothing in one number of the 
equation, it must do so in the other, that is; PE=Z must 
also become zero, and so must P’E, and we have instead of 

DE 
PE 
an expression which certainly amounts to nothing, unless we 
can show that the tangent of the angle DPE or PSB is 


Oo 
=2x simply one 


fe) 
equal to x 


Again it is impossible that a curve should ever be consid- 
ered as a polygon. The very definition of a curve is sim- 
ply that which distinguishes it from a straight line. Had 
the ancient geometers been willing to admit this principle, 
how easily could they have avoided the tedious and labori- 
ous “ reductio ad absurdum.” But they were too conscien- 
tious and exact to admit the possibility of establishing truth 
by even a doubtful principle. ‘In more modern times the 
greatest mathematicians and philosophers have, indeed, 
emphatically condemned the notion, that a curve is or can, 
be made up of right lines, however small. Berkeley, the 
celebrated Bishop of Cloyne, and his great antagonist, Mac- 
laurin, both unite in rejecting this notion as false and unten- 
able. Carnot, D’Alembert, Legrange, Cauchy, and a host 
of other illustrious mathematicians, deny that the circum- 
ference of a circle, or any other curve, can be identical with 


xX INTRODUCTION. 


the periphery of any polygon whatever.” So repugnant is 
this proposition to the fixed and fundamental conceptions of 
geometry, that it has been doubted and denied in all ages — 
by the most competent thinkers and judges. 

But notwithstanding all this, what shall we say when we 
find that the equation 

DE 
PE 2” 

is not merely an approximation to the truth, but that it is 
perfectly and exactly true. We have said that errors have 
manifestly been committed in arriving at this result. The 
advocates of the system point to the result and say behold 
the proof that we are right. The explanation of this seem- 
ing mystery was made long ago by Bishop Berkeley. “ For- 
asmuch,” says he, ‘‘as it may perhaps seem an unaccount- 
able paradox that mathematicians should deduce true prop- 
ositions from false principles, be right in the conclusion, and 
yet err in the premises, I shall endeavor particularly to 
explain why this may come to pass, and show how error may 
bring forth truth, though it cannot bring forth science.” He 
then proceeds to give an illustration, for which we will sub- 
stitute a similar one adapted to the figure we have chosen. 

It will be perceived that the curve we have taken is a par- 
abola whose axis is that of ordinates, and whose i aa 
is 1. Nowif we consider the curve PP’ as infinitely small 
and a straight line, the angle P’PE would be the Rar we 
are seeking; but since the curvecan never become a straight 
line, the increment P’E must always be too great, and the 
point P’ must always be above the tangent line; so that the 
error arising from taking PP’ as a part of the tangent line 
will be equal to P’D, or that part of the increment of the 
ordinate which lies between thetangent and the curve. But 
we have found 


mm 


yy —yH2xh+h* =P'E 


INTRODUCTION. xxl 


and since from the nature of the parabola 
x 
SB=— 
2 


we have from similar triangles 
DEaPH ra. 55 


or 
DE:2:: ete ces f 
SAiyi Zee el gy 
hence 
DE=2xh 
and 


P'D=P’E—DE=24h+h? —2xh=h® 
Here then we find that by throwing out 2” we exactly cor- 
rect the error arising from taking the curve PP’ asa straight 
line; and reduce the line P’E=y'—y to DE the line we are 


: DE est ec. 
seeking to fix the value of PE? which is the tangent of the 


angle which the tangent line makes with the axis of abscissas. 

Thus the infinitesimal method arrives at the true result, 
not because its principles are true, nor because its errors are 
small, but because they are, whether great or small, exactly 
equal, and exactly cancel and destroy cach other. This theory 
then is not only false but unnecessary. The false proposi- 
tions with which it sets out are as unnecessary as they are 
absurd. The errors may as well be great as small. The 
system is but a mere artifice, which “ by means of signs and 
symbols and false hypotheses, has been transformed into the 
sublime mystery of the transcendental analysis.” We dis- 
miss it, therefore, with the remark, that to admit and accept 
an error, even infinitesimal, in mathematics, strips the sci- 
ence of its chief glory, and introduces darkness and doubt 
where only the pure light of truth should prevail — in fact 
it opens the ‘door for anarchy in all science, and unsettles 
the very foundations of all knowledge. 


Xxil INTRODUCTION. 


THE METHOD OF LIMITS. 


The philosophical principle on which this method is 
based, is thus stated by Sir Isaac Newton, in the enunciation 
of the first lemma in the first book of the Principia. 

“ Quantities and ratios of quantities, which in any finite time 
converge continually to equality, and before the end of that time 
approach nearer the one to the other than by any given differ- 
ence, become ultimately equal.” 

The principle here stated would be applied to the solution 
-of our problem in the following manner. 


Th we BAW : f ; 
€ ratio PES considered as the ultimate value of the ratio 


A meee 
PE which approaches it as BB’ decreases, and coincides 


with it when B’ has reached the point B, for then the points 
U 


P’ and Dwillhavecome together. Again é ae =2x-+h ap- 


proaches the value of 2x as #, or B’B, diminishes, and 
reaches that value when 4 becomes zero. 
PE yy : 
Now since Soe and since by the lemma of New- 
DE 
ton >Re becomes ultimately equal to EE? and since by the 


same lemma = = becomes ultimately equal to 2x, at the 


: Dita 
same time and for the same reason, it follows’ that PE 3S 
equal to 2x. 
It will here be seen that the point on which this demon- 


, 


stration turns is, that since the ratio PE approaches the 


DE 2 . 
ratio pp? 3s BB’ decreases, nearer than by any given dif- 


INTRODUCTION. XXIil 


ference, they become, according to the lemma, ultimately 
PE y'—y DE 
equal; and the equation Si oy Oe becomes pr 2” 
which is its limit. . 
The objection that lies immediately under the surface of 
this demonstration is, that when BB’ or % has become zero, 


and we have for a result 
Date hs. ear DE 03 
A —=2x instea O PE —=2x 


Thus while we arrive at the desired value of our ratio, the 
ratio itself has lost all meaning, or at least all the attributes 
of quantity. 

The mind of Newton was too acute not to perceive the 
apparent absurdity involved in this application of his prin- 
ple, and he therefore gives the following explanation and 
defense of it: . 

“ Perhaps it may be objected that there is no ultimate pro- 
portion of evanescent quantities; because the proportion 
before the quantities have vanished is not ultimate, and 
when they have vanished is none. But by the same argu- 
ment it may be alleged, that a body arriving at a certain 
place, and there stopping, has no ultimate velocity; because 
the velocity before the body comes to the place is not its 
ultimate velocity; when it has arrived is none. But the 
answer is easy; for by the ultimate velocity is meant that 
with which the body is moved, neither before it arrives at its 
last place, and the motion ceases, nor after, but at the very 
instant it arrives: that is, the velocity with which it arrives 
at its last place, and with which the motion ceases. And in 
like manner, by the ultimate ratio of evanescent quantities, 
-is to be understood the ratio of the quantities, not before 
they vanish nor afterwards, but with which they vanish.” 

The illustration here given by Newton unfortunately 
throws no additional light upon the subject. It is certainly 


XXIV INTRODUCTION, 


no easier to conceive a body coming to a stop zw7/h any 
velocity, than it is to conceive of quantities vanishing wth 
aratio. Sir Isaac Newton is quite right when he says that 
the same argument may be alleged against both propositions, 
for both involve notions that are equally repugnant to our 
reason and consciousness; one that there may be a ratio 
without quantities, and the other that there may be velocity 
without motion. In fact the illustration is, if anything, the 
more objectionable notion of the two, for the very reason 
why a body stops in its motion is because its velocity has 
expired and is gone. 

The argument based on Newton’s lemma has been, by no 
means, universally received even among those mathemati- 
cians who reject the philosophy of Leibnitz. 

M. Comté, who views the method of limits with consider- 
able favor, says of it: “It is very far from offering such 
powerful resources for the’solution of problems as the infin- 
itesimal method. The obligation it imposes of never con- 
sidering the increments of magnitudes separately and by 
themselves, nor even by their ratios, retards considerably the 
operations of the mind in the formation of auxiliary equa- 
tions. We may even say that it greatly embarrasses. the 
purely analytical transformations.” 

Again in speaking of the course adopted by certain geom- 


dy 
eters, he says: “ In designating by 7 “. 7, that which logically 


ought, in the theory of limits, to be denoted by Lo (that 


Increment of ¥ 


is, Limit Tee i} and in extending to all the other 


rement of Xx 
analytical conceptions this displacement of signs, they in- 
tended, undoubtedly, to combine the special advantages of 
the two methods, but, in reality, they have only succeeded 
in causing a vicious confusion between them, a familiarity 


INTRODUCTION. XXV 


with which hinders the formation of clear and emict ideas 
of either.” ~ 

Says Lagrange: “That method has the great inconven- 
ience of considering quantities in the state in which they 
cease, so to speak, to be quantities; for although we can 
always well conceive the ratio of two quantities, as long as 
they remain finite, that ratio offers to the mind no clear and 
precise idea, as soon as its terms na¥e become the one and 
the other nothing at the same time.’ 

But the objection to the lemma in question as a funda- 
mental principle of the calculus lies deeper than in its weak- 
ness and inefficiency. The proposition carried to its legiti- 
mate results, overthrows the very system it 1s supposed to 
establish. Says the acute and candid Bishop Berkeley, in 
reply to Jurin: “ Fora fluxionist writing about momentums, 
to argue that quantities must be equal because they have no 
assignable difference, seems the most injudicious step that 
could be taken; it is directly demolishing the very doctririe 
you would defend. For it will thence follow that all hom- 
ogenous momentums are equal, and consequently the veloc- 
ities, mutations or fluxions proportional to these are likewise 
equal. There is, therefore, only one proportion of equality 
throughout which at once overthrows the whole system you 
undertake to defend.” 

This is conclusive. If quantities that during any finite 
time constantly approach each other, and before the end of 
that time approach nearer than any given difference are 


4 


xe | 
ultimately equal, then not only >> PE and, =; jz become ulti- 


Migtervacdttan butler, eb). bh P’B. and es all become 
ultimately equal, for they all fulfill the conditions required by 


D ° 
lemma. Hence instead of pr 2%, or even [=24, we 


have as the logical sequence of this proposition 2v=1, 


XXV1 INTRODUCTION. ‘ 


‘Although the method of limits has generally been attrib- 
uted to Sir Isaac Newton, who was the author of the prop- 
Osition which has served for its foundation, it is certain that 
this method as applied to the Differential Calculus, or method 
of fluxions, was not his. He has laid for that a very differ- 
ent foundation, as we shall see in due time, which has cer- 
tainly this merit, that there is nothing false nor absurd 
about it; and if it doesnot clearly unravel the mysteries of 
the calculus, it places in our hands the only clue by which 
we can do it for ourselves. 


METHOD OF LIMITS APPLIED TO THE DOCTRINE OF RATES. 


Prof. Loomis, of New Haven, has undoubtedly adopted 
the true conception of the nature of a differential. But he 
has unfortunately attempted to combine it with the method 
of limits, and has, therefore, become entangled in the same 
inconsistencies that we have already found to be inseparably 
attached to that method. 

It will be observed that in the following demonstration 
taken from his calculus, the question hinges on the ratio of 
the differential or rate of change of a variable to that of its 
square. And hence the demonstration although not arising 
directly in a question of tangency, must yet be tested by 
the same principles as those we have examined. 

“Tf the side of a square be represented by ~ its area will 
be represented by x?. When the side of the square is 
increased by 4, and become x+%, the area will become 
(x+/)?, which is equal to 

x*+2xh+h* : 
While the side has increased by 4, the area has increased by 
2xh+h?, If then we employ 2 to denote the rate at which 


x increases, 2x2+h* would have denoted the rate at which > 


a 


INTRODUCTION. eva 


the area increased had that rate been uniform; in which case 
we should have had the following proportion, 
rate of increase of the side:rate of increase of the area 
2:2: 2xh +h? ::1:2x+/ 
but since the area of the square increases each instant more 
and more rapidly, the quantity 2x+4 is greater than the 
increment which would have resulted had that rate been 
uniform; and the smaller Z is supposed to be, the nearer does 
the increment 2x+ approach to that which would have 
resulted, had the rate at which the square was increasing, 
when its side became x, continued uniform. When Zis equal 
to zerv this ratio becomes that of 
20 

which 1s, therefore, the ratio of the rate of increase of the 
side to that of the area of a square when the side is equal 
tie 

This demonstration is plausible, but will not bear close 
examination. We are told that when Z is equal to zero, the 
ratio of the rate of increase of the side of the square to that 
of its area is that of 1 to 2x. But by hypothesis / repre- 
sents the rate of increase of the side of the square; if then 
A becomes zero, the side (and of course the area) of the 
square wel have no rate of increase, and hence instead of 
rate of increase of the side: rate of increase of thearea:: 1: 2% 
we shall have 

O:O0::1: 2% or f=2x 

a result that we are already familiar with. 


A COMBINATION OF METHODS. 


Another text-book of very high authority sets forth the 
following method, which seems to be partly derived from 
Lagranye’s idea of derived functions. We give the author’s 
explanation in his own words: 


XXVI1 INTRODUCTION, 


“ To explain what is meant by the differential of a quantity 
or function, let us take the simple expression 
u=ax® (1) 
in which z is a function of x. Suppose x to be increased by 
another variable “%; the original function then becomes 
a(x+h)* ; calling the new state of the function #’ we have 
uw’ =a(x+h)*=ax? +2axh+ah? 
From this subtracting equation (1) member from member 
we have 
u —u=2axh+ah? 
The second member of this equation is the difference be- 
tween the primitive and new state of the function ax, while 
his the difference between the two corresponding states of 
the independent variable x. As &% is entirely arbitrary, an 
infinite number of values may be assigned to it. Let one 
of these values, zich zs to remain the same while x ts inde- 
pendent, be denoted by dx, and called differential of x, to 
distinguish it from all other values of %. This particular 
value being substituted in equation (1) gives for the corres- 
ponding difference between the two states of w or x* 
uw’ —u=2ax dx+ta(dx)? (2) 
Now the first term of this particular difference 7s called the dif- 
JSerential of u, and is written 
du=2ax .ax 


The coefficient (2ax) of the differential of x, in this expres- 
sion, is called the differential coefficient of the function u, and 
is evidently obtained by dividing the differential of the func- 
tion by the. differential of the variable, and 1s in general 
written ” 
oe eax (3) 
Now will any inquiring student be satisfied with this 
“explanation?” Will he infer from it anything of the nature 


INTRODUCTION, XX1xX 


and office of a differential, and what is its philosophical 
relation to the function to which it belongs ? 

But perhaps this is not intended as a full explanation, for 
the author proceeds: 

“ Resuming this expression 

ui —u=2axh+ah* 
and dividing by 2, we have 

wu! —U 

h 

In the first member of this equation the denominator is the 
variable increment of the variable x, and the numerator is 
the corresponding increment of the function w,; the second 
member is then the value of the ratio of these two incre- 
ments. As # is diminished, this value diminishes and be- 
comes nearer and nearer equal to 2ax, and finally when =o 
it becomes equal to 22x. From this we see, that as these 
increments decrease, their ratio approaches nearer and nearer 
to the expression 2ax, and that by: giving to 4 very small 
values, this ratio may be made to differ from 2ax by as small 
a quantity as we please. This expression is then properly, 
the limit of this ratio, and is at once obtained from the value 
of the ratio by making the increment =o. It will also be 
seen that this limit is precisely the same expression as the one 
which we have called the differential coefficient of the func-’ 
tion wz.” 

Now we have here a term arbitrarily selected without 
explanation or apparent reason, in which /% has a fixed value; 
and this term is called the differential of the function. But 
afterwards in order to find the value of this term it becomes 
necessary to reduce % to zero. The question arises here, if 
h must be zero in the one case why must it not be zero in the 
other? It will not answer to say that 2ax, the differential 
coefficient, does not contain # or dx, and is therefore not 
affected by its value; because @x des occur in the first mem- 


=2ax+ah 


Xxx INTRODUCTION. 


ber of equation (3) as its denominator, and hence as to do 


: Lu d 
with the value of 2ax. We have then 2ax= Ts where ¢x 1s 


} u—tu 2 
a particular value of 4, and 2ax =F where Z is equal to 


zero. If the case be so that the value of the fraction 
(Li hat, ed aad : 

Tz OF 7% 18 independent of that of the denominator, the 
author nowhere tells us why it is so. 

The object of the author in this setting forth of his method 
is evidently to avoid the apparent absurdity of making @v=o 
while it performs so important an office in his subsequent 
analysis; thus escaping one of the inconsistencies of the 
method of limits. But as he practically uses that method in 
the first part of his work, there seems to be after all some 
confusion of ideas, and it is difficult to regard dx as having 
a fixed value, while its representative 2 is continually re- 
duced to zero. The author himself seems to have become 
. wearied with this indirect and misty method, and when he 
comes to the practical application of the Calculus to Geom- 
etry, comes squarely down on to the infinitesimal system 
which, with all its inconsistencies, is far more direct and 
fruitful in its results than the method of limits. 


The most striking circumstance in connection with every - 


modification of the method of limits is, that the value of the 
differential coefficient is the objective point sought after. 
Whether this is obtained by a sound, logical demonstration ° 
or not, it is the only thing found which has a real value, of 
which we can form a definite conception. Now the differ- 
ential itself is quite as important as the differential coeffi- 
cient. It is true we do not regard its actual measured value, 
but we do regard its relative value as compared with that of 
other differentials; and for this purpose we need some value 
that the mind can grasp and upon which the imagination can 


a a 


INTRODUCTION. XXxXI 


rest with satisfaction. But none of the systems we have 
examined present the idea of a differential as consisting of 
any such quantity. 

The infinitesimal system does indeed profess to give a sort 
of value to the differential. It is the “last value of ‘the 
variable before it becomes zero,” or “ the difference between 
two consecutive values of the variable ”— words that con- 
vey no more definite idea of quantity than a sort of attenu- 
ated essence of one about to vanish; while the system of 
limits leaves us nothing whatever but the ghosts of those 
that have departed entirely. 

Who would not desire to be relieved from the constant 
strain upon the imagination, and the severe draft on the 
faith required to attain the results of the calculus by such 
feeble means? What a relief to have placed in our grasp a 
principle that has substance and vitality; that is adapted to 
our conceptions and meets the demands of our reason, whose 
meaning is not dim nor doubtful, but clear as the noonday 
sun, shining by the light of its own self-evident truth. 

We have said that a fundamental error runs through all 
the systems of infinitesimals and limits, arising from the 
method of approaching the subject. To understand clearly 
what this error is we must have a clear conception of the 
true nature of a differential, and of the symbol which rep- 
resents it. As we have already stated, it is truly defined by 
Prof. Loomis as “ the rate of variation of a function or of any 
variable guantity,’ and further, “by the rate of increase at 
any instant we understand what would have been the absolute 
increase if this increase had been uniform.” And the dif- 
ferential coefficient is the ratio between the rate of varia- 
tion of any variable and the consequent rate of variation of 
the function into which it enters. Now this ratio is really 
what is sought after in both the systems that we have exam- 
ined. In the system of Leibnitz the infinitesimal increments 


“~ 


XXXil INTRODUCTION. 


represent the rates of increase, and in the method of limits 
the ratio of the rates is obtained from that of the actual 
increments by reducing them to their vanishing point. Not 
that the authors of these systems were conscious of any 
such meaning in their methods, but this was, nevertheless, 
the real, though unrecognized, philosophy on which those 
methods were based. Now the error which gaye rise to all 
the absurdities, sophisms and obscurities of their system was 
this — “hey endeavored to arrive at the ratio of the rates of 
change by means of the actual changes. ‘That is, they gave to 
the variable an increment, and to the function a correspond- 
ing one, and from these attempted to derive what is really the 
ratio of the rates, or the differential coefficient. This can- 
not logically be done, except in the case of uniform varia- 
tion ; for in all other cases the rate changes as the value of 
the function changes; so that before the rate can be meas- 
ured by any actual change, it will itself have changed. 
Take a familiar illustration. It is a well-established fact in 
Natural Philosophy, that the velocity of a body falling 7x 
vacuo to the earth cannot possibly be measured for any one 
instant by the actual movement of the body subsequent to 
that instant, for no such subsequent movement will be made 
with the same velocity. Now if no actual change can rep- 
resent a variable rate of change, the ratio of the actual 
changes cannot truly represent the ratio of the rates how- 
ever small they may be made. 

It is this effort to do what is, in the nature of things, im- 
possible, that has introduced all the difficulties, enigmas and 
mysteries that have beset the differential calculus from the 
beginning. Now these are as unnecessary as they are 
objectionable. The true principles of the science are as 
clear and consonant with reason as the elements of Euclid, 
and the science itself flows from them as directly as the 
light from the sun. Not only so, but while the methods we 


INTRODUCTION, XXXIll | 


have examined have produced a study hard and unattractive, 
consisting almost entirely of manipulations of the mere 
machinery of analysis, the subject is really full of beauty, 
abounding in ideas of the most novel and interesting kind, 
and furnishing a field for the exercise of the imagination 
that will tax all its powers —it is, in fact, the poetry of 
mathematics. 


THE TRUE METHOD OF NEWTON. 


The method of arriving at the differential coefficient by 
means of the ultimate ratios of the increments, or, in other 
words, the method of limits, has generally been ascribed to 
Sir Isaac Newton; but this is evidently an error. The 
theory on which that method is founded is certainly his, and 
it is but just that he should be held responsible for the re- 
sults that legitimately flow from it. But it is not the theory 
on which he formed 4zs method of fluxions. Zhafis con- 
tained in the second lemma of the second book of his Prin- 
cipia. In a scholium to that lemma he says: “In a letter 
of mine to Mr. J. Collins, dated Dec. 10, 1672, having 
described a method of tangents——which at that time was 
made public, I subjoined these words. TZhes zs one particular 
or rather corollary, of a general method, which extends itself, 
without any troublesome calculation, not only to the drawing of 
tangents to any curved lines, whether geometrical or mechanical, 
or any how resolving other abstruse kinds of problems about the 
crookedness |curvature| areas, lengths, centers of gravity of 
curves, etc., nor ts tt limited to equations which are free fron 
surad quantities. This method I have interwoven with that other 
of working equations, by reducing them to infinite series. So 
far that letter. And these last words relate to a treatise I 
composed on that subject in the year 1671. The founda- 


XXXIV INTRODUCTION. 


tion of that general method is contained in the preceding 
lemma.” 

Here it is distinctly stated by Newton himself that he had 
invented a general method which was applicable not only to 
the drawing of tangents, but to all the higher and more del- 
icate problems which appear in the differential calculus, and 
that this general method has dhe lemma in question for tts 
FOUNDATION. 

We have then but to examine this lemma to ascertain the 
real basis on which the “method of Newton” was con- 
structed. Forthis purpose we give the lemma in the author’s 
own words. 


LEMMA II. 


“ The moment of any genttum ts equal to the moments of each 
of the generating sides drawn into the indices of the powers of 
those sides, and into thetr coefficients continually. 

“T call any quantity a genitum which is not made by the 
addition or subduction of divers parts, but is generated or 
produced in arithmetic by the multiplication, division or ex- 
traction of the root of any terms whatsoever; in geometry 
by the invention of contents and sides, or the extremes and 
means of proportionals. Quantities of this kind are pro- 
ducts, quotients, roots, rectangles squares, cubes, square 
and cubic sides and the like. 

“These quantities I here consider as variable and inde- 
termined, and increasing or decreasing as it were by a per- 
petual motion or flux; and I understand their momentane- 
ous increments or decrements by the name of moments; so 
that the increments may be esteemed as additive or affirm- 
ative moments, and the decrements as subducted or nega- 
tive ones. But take care not to look upon finite particles as 
such. Finite particles are not moments, but the very quan- 
tities generated by the moments, We are to conceive them 


INTRODUCTION. XXXV 


as the just nascent principles of finite magnitudes. Nor do 
we in this lemma regard the magnitudes of the moments, 
but their first proportion as nascent. It will be the same 
thing, if, instead of moments, we use either the velocities of 
the increments and decrements (which may be called the 
motions, mutations and fluxions of quantities), or any finite 
quantities proportional to those velocities. The coefficient 
of any generating side is the quantity which arises by 
applying the genitum to that side. 

“Wherefore the sense of the lemma is, that if the mo- 
ments of any quantities A, B, C, etc., increasing or decreas- 
ing by a perpetual flux or the velocities of the mutations 
which are proportional to them, be called a, 4, c¢, etc., the 
moment or mutation of the generated rectangle AB will be 
aB+6A; the moment of the generated content ABC will be 
aBC+/AC+cAB; and the moments of the generated pow- 
ers A®, AS, At, AP, A? AS AS A-1, A-*, A~® will be 
— tA, *, aA? —ah—*, 


al 
2aA, 3aA*, 4aA%, 4aA *, 3aA® 


: PES. : ; 
—2aA~*,>—taA * respectively; and in general that the 


7D nm 
moment of any power A” will be “aA -m Also that the 
moment of the generated quantity A?B will be 2aAB+dA®; 
the moment of the generated quantity A®B4C? will be 
3aA*B*tC? +-40A3 BC? + 2cA2B4AC ; and the moment of the 
3 
generated quantity = or A®B-?, will be 3¢A*B-*— 
26A°B-%,andsoon. The lemma is thus demonstrated. 
“Case 1. Any rectangle, as AB, augmented by a perpet- 
ual flux, when as yet there wanted of both sides A and B, 
half the moments $a and 44, was A—4a into B—4é, or 
AB—4aB—46A+4ad ; but as soon as the sides A and B are 
augmented by the other half moments, the rectangle be- 
comes A+-}a into B+34, or AB-+-4aB+30A+ j0d, From 


XXXVi INTRODUCTION. 


this rectangle subduct the former rectangle, and there re- 
mains the excess a@B+0A. ‘Therefore with the whole incre- 
ments a and & of the sides, the increment aB+0A of the 
rectangle is generated Q. E. D.” 

“Case 2. Suppose AB always equal to G, and then the 
moment of the constant ABC or GC (by case 1) will be 
gC+cG, that is (putting AB and aB+éA for G and g) 
aBC+é4AC+cAB. And the reasoning is the same for con- 
tents under ever so many sides. Q. E. D.” 

It is unnecessary to quote the demonstrations of the other 
cases, as they all flow naturally and logically from these which 
form the key to the whole system. 

We must concede that this demonstration is not as clear 
and complete as could be desired. Let us, however, 
endeavor to extract from it the real, though perhaps some- 
what vague conception of the subject which occupied the 
mind of Newton. It is to be remarked, however, that the 
doctrine of “mts is nowhere hinted at, but the results are 
direct, positive and substantial. 

The first question suggested by the lemma is, what is 
really meant by the term “ moment.” It might at first seem 
that the “ moments” of Newton were in fact the same thing 
as the differentials of Leibnitz, for he speaks of them as 
something (though not finite quantities) to be added or sub- 
tracted. But a very little examination of the lemma will 
dispel the notion. Their magnitudes are not to be regarded. 
But the magnitudes of the differentials of Leibnitz are to be 
regarded as infinitely small. Again, “finite particles ” are 
not “moments,” but the “very quantities generated by the 
moments.” Now the differentials of Leibnitz never gener- 
ate anything; they are the infinitesimal remains of incre- 
ments that have been added and then taken away. Again, 
moments are the “nascent principles of finite magnitudes.” 
But the “principles” which generate “ finite magnitudes ” 


INTRODUCTION. XXXVII 


or increments can be nothing else than the Zaws which con- 
trol the changes in the “ genitum;” that is, THE RATE OF 
CHANGE. This interpretation is confirmed by the further 
statement that we may use instead of them “the velocities ” 
or any finite quantities proportional thereto. Hence we 
infer that a, 4, c, which are called moments, are intended as 
“ symbols to represent the rates of change, being finite quan- 
tities proportional to those rates, and as the quantities 
A, B, C, etc., are increasing or decreasing by a “ perpetual 
flux,” that is by a uniform rate of change, the actual incre- 
ments or decrements a, 4, ¢ will represent those rates. So 
that the difference between A—4a and A+4a (equal to a) 
represents the rate of increase of A, and the difference be- 
tween B—46 and B+44é (equal to 4) accruing during the 
same time represents the corresponding rate of increase of 
B; and the ratio of a to 4 represents the ratio of those rates 
whatever may be their magnitude as symbols. But while 
these symbols or suppositive increments (being produced 
at a uniform rate) represent the respective rates of increase 
of A and B, we are told that the corresponding increment 
of their product (2B+<A) represents the “moment” or rate 
of increase of their product. Now as the product does not 
increase at a uniform rate, it becomes a question why Z/és 
increment should represent the raze of increase of AB. This 
is probably one of those cases in which the intuitive per- 
ceptions of Newton seized the true result without stopping 
to elaborate the intermediate steps. At all events he has 
here presented the only true key that will completely unlock 
the calculus; and this key we shall in due time apply to that 
purpose. 


XXXVII1 INTRODUCTION, 


[The following letter from the author to a correspondent contains a 
more elaborate explanation of some of the principies of the calculus than 
will be found in the text of this work. It was written to meet certain 
objections, which will appear in the progress of the letter, and is inserted 
here in order to meet the same objections should they arise in the minds 
of others. ] 


CHIC aco, Jans227.1375; 

DEAR Sir: You say that you are in doubt as to what 
meaning I attach to the statement: “ This law is derived, 
not from any actual change, but from the conditions con- 
tained in the algebraic formula by which the variable func- 
tion is expressed.”” My meaning is, that in any variable 
function, as for instance ax—dx* the rate of variation, or, in 
other words, the /aw of change in the function is to be de- 
rived, zo¢ from giving an actual increment to x and a corres- 
ponding one to the function, but from the conditions expressed 
in the formula; that is from the expressed relation of x to 
its function; and the expression of this law, derived from the 
formula is — the change that WOULD take place in the function, 
arising from the rate of change in x, tf the change in the func- 
fon were to continue uniform. ‘To find this suppositive uni- 
form change in the function is one of the problems of this 
work. We willtake however a very simple case for illustra- 
tion. Suppose we have the function ax in whichx is chang- 
ing at a variable rate. The law of change in this function 
is derived from the simple condition expressed in the for- 
mula, viz., that as the value of the function is always atimes 
that of x, the rate of change in the function will always be 
a times that of «; and whatever uniform change may be 
given to x to express its rate of change, the corresponding 
uniform change in the function will be a times that of x and 


, 


INTRODUCTION. XXX1X 


this is the “law of change derived from the conditions con- 
tained in the algebraic formula ax’? — hence d(ax)=adx. 

Having thus expressed my meaning, permit me to reply 
to some expressions in your letter which I will quote out of 
their connection, but not, I think, so as to change their 
meaning. You say: “The amount of change in any par- 
ticular case will be determined by the law,” etc. I under- 
stand you to mean that the amount of change for any unii 
of time will be determined by the law governing the change 
—that is, by the rate of change. This is true for a uniform 
change but not for a variable one. For, suppose two quan- 
tities are changing, at a variable rate, in such a manner that 
at a certain instant, the rate of each is the same; then the 
law of change at ¢hat instant is the same in both, while the 
actual amount of change zz any unit of time in each quantity 
may be as different as possible from the other. Hence the 
amount of a variable change will zo¢ be determined for any 
unit of time by the law or rate of change. 

Again you say, in reference to my statement already 
quoted: “I see no occasion for the remark referred to unless 
it mean that there is no sch actual change as the supposi- 
tive change symbolized by dx; a very different statement 
from the one made.” 

Now it seems to me that we are using the same word in 
different senses. I mean by a swffosttive change in the case 
of a variable rate, an zdea/, or (if you please) fictitious or 
hypothetical one, which is not only different from any actual 
change but different from what any actual change can be. 
Hence then, of course, there can be “no sch actual change 
as the suppositive one symbolized by dv ;” and I have re- 
peatedly stated that the. change symbolized by @v was not 
an actual change a/a//. If, for example, I say that a body 
is falling freely to the earth at the rate of fifty feet in one 
second, this fifty feet is wholly an zdea/ change of position, 


x] INTRODUCTION. 


or increment of space; and I would never be understood as 
asserting that it was an actual increment of the space passed 
over by the falling body in one second; nor that any fifty 
feet would be uniformly described; and yet my assertion 
would mean fifty feet passed over at a uniform rate; but it 
would be a “ suppositive’’ fifty feet, and would be passed over 
in one second on ¢he supposition that the rate continued uni- 
form for that time: but that never does or can take place in 
reality. 

Again you say: “ Admitting your idea of variables as in 
a state of change, it still seems to me that in the calculus, 
we are concerned with the changes which result from that 
state, zo¢ with the state itself; that zx is a symbol not of a 
state, but of a certain amount of change that would accord- 
ing to your definition occur in a unit of time, were the 
change at a particular point, or rather any point continued 
uniformly for that unit, and that if @ is an entity there must 
be an actual change as its basis.” 

Now in the calculus it is exactly the state of change, and 
not the actual change with which we are concerned. The 
latter belongs more properly to Natural Philosophy. It is 
true that dx is the symbol of a certain amount of supposi- 
tive change (not real except the rate be uniform), and this 
symbol expresses the state of change, and not an actual nor 
(where the rate is variable) even a possible one. But be- 
cause the change is ideal, suppositive or hypothetical, and 
not actual nor possible, it does not follow that it is not an 
entity. ‘The ideal change is the symbol of the rate, and the 
rate though not a simple quantity, but a ve/ation between two 
other quantities (viz., change, ard the time occupied by it), 
and, therefore, represented by a symbol is still an evdity, for 
anything that exists is an an entity. And although rate is 
not change, it does not follow that it is based on “no 
change;”’ it is something wholly different from change, 
although represented by it as a symbol. 


INTRODUCTION xi 


You ask in reference to signs: “ Do we not in Trigonom- 
etry obtain our general formulas as though the functions of 
the angle were all plus, and when applying them make such 
changes as the signs of particular values require ?”’ 

In Trigonometry we consider all the functions as essenti- 
ally positive in the first quadrant, but outside of that some 
of them are essentially negative, whatever may be the szgz 
prefixed. We must distinguish in the calculus between the 
signs plus and minus, and the ¢erms positive and negative. 
The former denote the ve/aton of one quantity to another, 
the latter the essential character of the quantity. 

I now come to the last point mentioned in your letter. It 
is a vital one, and I wish in discussing it to make my argu- 
ment such, that taken in connection with the demonstration 
in the book itself, it will be complete and exhaustive. 
Hence I shall be obliged to repeat some things that I have 
said in order that the continuity of the argument may not 
be broken. 

You say, “I do concede that you have logically obtained 
the differential of the product of two variables upon the 
assumed hypothesis, viz., that the variables changed uniformly ; 
that if they do not change uniformly, then by the method 
you have adopted dz would not be the change in ~ corres- 
ponding to contemporary states of x and y for x and y would 
not be contemporaneous.” 

The question, therefore, is, will the demonstration which 
I have given, and which you concede to be logical (and, of 
course, conclusive) for obtaining the differential of a pro- 
duct of two variables when their rate of change is constant, 
be also true when their rate of change is variable ? 

First. What do we mean by rade of change? Rateisa 
complex idea. Its elements are change, and the “me con- 
sumed by it. It is, in fact, the relation between these two. 
It is an entity because it exists. It is a quantity because it 


xlil INTRODUCTION 


can be increased, diminished and measured; but itis a very 
different kind of quantity from either time orchange. These 
latter are quantities of which the idea is oljectve, presented 
externally to the mind; while va is a quantity of which 
the idea is subjective, originating zz the mind and only capa- 
ble of being represented by a symbol. This symbol is of 
course arbitrary. Sometimes it is the time (the change be- 
ing known) which increases as the rate diminishes and we 
versa. But in the Calculus the symbol is the eange, which 
increases and decreases with the rate; and as this symbol is 
only used to compare one rate with anather, we need not 
regard the element of time farther than to make the symbol- 
ical changes simultaneous. By thus leaving out the element 
of time the idea of rate has become so far identified with 
that of its symbol, that we often fail to make any distinction 
between them; but they are as different from each other as 
an angle from its sine or tangent. 

Second. The change required to be the symbol of a rate 
must be a wuzform one. Thus, if the rate of a falling body 
is said to be fifty feet in one second, it is meant that it would 
pass over that space in one second of time with a uniform 
motion tf the existing rate were not disturbed for that time. 
No change whatever except a uniform one caz be a symbol 
by which a rate will be correctly expressed. 

Third. To apply the idea of rate to an actual change 
which we will suppose to be a variable one, let us again take 
the case of a falling body, which at the moment in question 
has a velocity of fifty feet in one second. Then rate being, 
as I have said, the relation between time and a uniform 
change, it follows that in this case the rate is the relation of 
one second of time to fifty feet of space uniformly described 
in that time, and this relation is the Zaw of the motion at 
that instant, and unless interrupted by some external cause 
would continue to govern the motion indefinitely. Now in 


— 


INTRODUCTION. xiii 


this case, this law applies to no other instant of time nor 
point of space. For suppose a body to be projected upward 
with such a velocity that at the moment when the velocity 
of the falling body had increased to fifty feet in one second, 
that of the rising body should have diminished to the same 
rate. Then at “at moment the law governing the motions of 
both these bodies would be exactly the same. If the bodies 
are equal in weight and should both strike an obstacle, the 
effect of the impact would be the same in both cases, and 
also the same as tf their motions were uniformat that rate ; and 
yet they do not move at the same rate during any time at all. 
The existence of a rate then in a variable change is zwszan- 
tancous ; it occupies no time; no actual change takes place 
in accordance with the rate; and yet it is the same, its effects 
are the same, and 7z#s relations are the same ad the instant as 
though it were not changing at all. 

Fourth. (n obtaining then the rate of change in a pro- 
duct of two variables, we have only to consider their rates 
as they exist at chat instant, and the result will be the same 
whether those rates are constant or not; for we are not con- 
cerned with the rate at any other time either before or after 
that instant. 

It may be asked if a rate occupies no time and involves 
no actual unirorm change, why do I make use of such uni- 
form changes in obtaining the rate of a product? I reply, 
the uniform changes used for this purpose are not real, but 
suppositive or fictitious changes, which are symbols of the 
rates existing at the moment in the variables whose product 
is under investigation. The use therefore of such uniform 
suppositive changes in obtaining the rate of the product is 
not at all inconsistent with the proposition that the rate in a 
variable change does not require time non actual uniform 
change for its existence, (viz, its occurrence ortaking place), 

fifth. J have said that the change which symbolizes the 


xliv INTRODUCTION 


rate of change, whether that rate be uniform or not, masz be 
made at a uniform rate. ‘To show this practically let us take 
the product xy, in which we will suppose the rates of the 
factors are uniform, and represented by the uniform incre- 
ment zx and dy, made in the same unit of time. You con- 
cede that in this case I have logically obtained the result 
d(xy)=ydx+xdy, which is of course true for all values of x 
andy. Let us take some ove of these values, which we will 
consider to be the value of the product af the moment of dif- 
Serentiation. Since gx is a uniform suppositive increment of 
x,and y has its real value as it exists at the moment, the pro- 
duct yd¢x will be a suppositive wzz/orm increment of the fro- 
duct arising from the rate of x and proportional to it— y hav- 
ing undergone no veal change during the momentary exist- 
ence of the rate. Similarly «zy will be also a suppositive 
uniform increment arising from the rate of y ; and therefore 
their sum is the suppositive uniform change in xy arising 
from the rate of both x and y, and expresses the rate of 
change in xy at the instant of differentiation. Now the 
symbol or measure of the rate of change in x, y and’ xy 
being in each case a uniform change, it follows that when 
u(=xy) becomes one of the factors, the same reasoning will 
apply to it as to x and y separately, for it is the measure of 
the rate that I seek in both cases and the measures are unt- 
Jorm changes. 

You will notice in the demonstration in the book that I state 
distinctly that I suppose the “ variables A and B to be in- 
creasing at any rate whatever, either uniform or variable, 
and independent of each other.” My object was to find the 
rate of increase of their product when they had arrived at 
the two specific values of x and y. It is “zs point and no 
other that I consider the actwa/ values of A and B. It is at 
this point and no other that I consider their rates of increase. 
The symbols ¢x and dy, which are the suppositive or ficti- 


INTRODUCTION. xlv 


tious changes in A and B, refer to the rates at the one znstant 
when their values had reached the fixed points x and y, and 
to no other either before or after; for at any other, either 
before or after, their rates might not be the same. You will 
observe that I state expressly in the demonstration that 
“these suppositive increments are what we have to con- 
sider,” and I make no change in the real values of A and B. 
It must not be forgotten that what we obtain in the Calculus 
as the differential of a function is only the formula from 
which the true value is obtained by substituting the actual 
values of the variables, and of their rates of change as they 
exist at the moment of differentiation. To illustrate this let 
us take the following problem. Suppose a body to fall say 
from the moon to theearth. The force of gravity, or weight 
of the body, as well as the velocity, will constantly increase 
at an increasing rate during the fall. Suppose at some in- 
stant the weight of the body is fifty pounds and is increasing 
at the rate of two-hundredths of a pound in one minute, 
while at the same moment the velocity is one thousand feet 
and increasing at the rate of five feet in one minute. The 
problem is to find the rate of increase of the momentum. 
Call the weight x and the velocity y ; then 
X=50, y=1000, @x=.02, Ay=5 
ana the momentum xy=50,000. Substituting these values 
in the formula we have 
a (xy) =ydx+xdy=20+250=270 

Hence at ¢hat cnstan¢ the momentum is increasing at the rate 
of 324, or .54 per cent of its value in one minute; and if 
the rate of increase were to continue uniform it would in- 
crease that much in one minnte. But the rate is not uni- 
form, the momentum does not increase at all at that rate 
during any time, « and y make no actual change at the rates 
assigned to them; and yet dx, dy and @(xy) are entities and 
quantities, suppositive indeed, but which the imagination can 


xlvi INTRODUCTION. 


grasp and the mind comprehend; and it is such as these 
that I propose to substitute for the incomprehensible infini- 
tessimal or the equally incomprehensible quotient of nothing 
divided by nothing. Y ain, ete; 


The following outline of the argument in the foregoing 
letter may be of service in drawing attention to the chain of 
reasoning : 

first. A differential is a rate of change. 

Second. A rate is a subjective idea and can only be ex- 
pressed by a symbol. 

Third. ‘That symbol in the Calculus must be a uniform 
change, which, in a variable rate, is a fictitious one. 

fourth, Ina variable rate one such symbol can apply to 
but one value,and hence in sucha rade that one value requires no 
appreciable time for its existence nor ac/wal/ uniform change. 

Llifth. In comparing rates we compare their symbols, and 
in estimating them we estimate the symbols. 

Sixth. Hence allthe relations, effects and values of rates 
are at any one moment the same as if they were uniform. 

Seventh. Hence a demonstration applicable to uniform 
rates wil apply at azy one moment to variable rates also, for 
the symbols, which alone are used in the demonstration, and 
by which the-rates are expressed, compared and estimated, 
are precisely the same uniform changes, whether the rates 
are uniform or variable. 

Eighth. Vf a variable is changing at a variable rate, 
while that fact will not affect the expression for the differen- 
tial of its function, it will very sensibly affect the differential 
of the rate of the function, which is its second differential; 
and it is here and not in the first differential that we are to 


consider whether the rate of the variable is uniform or not.. 


Thus while a&xy) is always ydx+-xdy, yet if dy is variable, 
a xy=2dxdy+xd*y instead of 2dxdy as when the rates of 
both are constant. 


; 
Se 


ervelver lane Le 


DIFFERENTIAL CALCULUS. 


-DIFFERENTIAL CALCULUS. 


SL Ope EO Nek, 
_. DEFINITIONS AND FIRST PRINCIPLES. 


VARIABLES, 


(1) Two classes of quantities are considered in the dif- 
ferential calculus, namely, varzables and constants. 

Variables are quantities that are in a state of change ; that 
is, their values are in an increasing or decreasing condition ; 
such, for example, as the quantity of waterin a vessel which 
is being filled or emptied by a continuous stream; or as the 
force of attraction which increases or diminishes as the 
attracting bodies approach or recede from each other; or 
as the space between these same bodies while they are mov- 
ing. They are, in the differential calculus, not merely quan- 
tities subject to change, or to which different values may be 
assigned ; but quantities in which che change is supposed to be 
actually occurring at the moment when they become the sub- 
ject of the analysis. Itis their actual condition and not their 
attributes or qualities that are referred to in this definition. 
Take for example the space passed over by a falling body. 
That space is a variable, not because it may or does have 
different values, but because its value is constantly chang- 

4 


' 


50 DIFFERENTIAL CALCULUS. 

zg, oris in a state of change. It is this s¢aée, and not any 
actual change, that is the peculiar subject of the transcen- 
dental analysis. 


RATE OF VARIATION. 


(2) Rate of variation is the relation between the change of 
a variable and the #me occupied by the change. Being a 
relation and nota simple quantity, it can only be represented 
by a symbol, which is, a uniform change ina given unit of time. 
If the rate is constant, then the actual change is the true sym- 
bol, but if it is variable, then the change must be a supposz- 
tive one—that is, one that would take place in the same unit of 
time tf tt were to continue uniform. Thusin the case of a fall- 
ing body; it is said, the velocity at a certain moment is so 
many feet in one second. Itis not meant that the body actu- 
ally falls through that distance in one second, nor any dis- 
tance whatever at that rate, but that it would fall so far if the 
velocity. existing at that moment were to continue uniform for 
one second. The velocity belongs to that one moment, and 
that one position only. At the very next point above and 
below this position the velocity is different, and hence no 
actual ;novement, however small in respect to space and time, 
can possibly represent it. This will be seen at once if we 
consider what velocity is. It is not of itself a quantity, but 
a relation, which refers, not to the place, but to the condition 
of the body in respect to the motion —that is, to the degree 
or zutensity of the state of motion in which the body is. 

So it is with all variables. The rate of change refers to the 
intensity with which the change is going on, andif it is not 
uniform it cannot possibly represent the rate, for it lacks the 
essential element of the required symbol. The latter must 
therefore be obtained from the Zaw which governs the change 
and not from the change itself. Hence instead of giving to a 
function an actual increment for the sake of obtaining its rate of 


DEFINITIONS AND FIRST PRINCIPLES. 51 


increase at any moment, we examine the /aw which governs 
the change; and the expression of this law is “e change that 
would take place ina unit of time tf the rate were lo continue 
uniform ; AND THIS IS THE MEASURE OF THE RATE. 


Note. —It must be remarked that the ides of time, motion and velocity, attached 
to the ordinary meaning of these words, have no place in the abstract science of the 


differential calculus. The term ot7oxn and velocity are used in this article merely to 


ae ” 


illustrate the meaning of the term ‘‘7aze.”” It is true that velocity is a rate—the 
rate of motion. But many other things beside motion have a rate ; such as the varia- 
tion of light, heat, magnetism, force, anything which increases or diminishes by the 
operation of a prescribed law ; and the calculus is applicable to all such subjects where 
the conditions can be expressed analytically. 

The idea of z7ze in its absolute sense is also foreign to the calculus. The term 
‘* unit of time”’ in the definition does not refer to any specific portion of time, it may 
be great or small; its value does not enter into the calculation, and hence this system 
does not in any wise invade the domain of natural philosophy. All that the aéstract 
science of the calculus has to do with ¢Z7e, is confined to the simple condition that the 
suppositive changes in the value of the variable and its function, which symbolize their 
rates of change, shall be szszultaneous. And this is no more than all systems of the 
alculus require for the actual changes which are supposed to be made in the same 
quantities. 

In the application of the calculus to Geometry the idea of motion is in some sort 
introduced ; but not, however, in its philosophical sense as having an aésolute value. 
Geometrical magnitudes are supposed to be generated by-the movement of their ele- 
ments. Thusa line is generated by the flowing of a point, a surface by a line, anda 
solid by a surface ; and this conception is used to determine the proportion of magni- 
tudes, by comparing the rates at which they are generated instead of comparing the 
magnitudes themselves with each other. This idea of the generation of magnitudes 
by means of their elements is not new in mathematics. It is one of the seminal 
ideas of the Cartesian system; and though in this work it is certainly made more 


prominent than it has usually been, and more prolific in results, it is not therefore out ~ 


of place. 


DIFFERENTIALS. 


(3) Zhe differential of a variable, or function, ts tts rate of 
change or variation, symbolically expressed by the suppositive 
change that would take place at that rate. 

If the variable is essentially postive and increasing, or neg- 
ative and decreasing, its differential will be essentially fosz- 
tive. If it is essentially negative and ¢ucreasing, or posttive 
and decreasing, its differential will be essentially negative. 
Thus, if we consider a northern latitude positive and a 


52 DIFFERENTIAL CALCULUS. 


southern negative, a vessel will have a positive rate (or dif- 
ferential) of progress if her northern latitude is increasing 
or her southern latitude is decreasing; and wice versa her 
rate of progress will be negative. 

The xotaton used to designate the rate of variation or dif- 
ferential is the letter @ placed before the variable, whose 
rate is required. Thus the differential of x is written dx. 
If the variable is a component expression such as x*+ay, 
the differential would be written ¢ (x?+ay). Variables are 
themselves indicated by the last letters of the alphabet. 


Norte.—I use the nomenclature and notation of Leibnitz, not because there is any 
actual zecessity for so doing, but because their use has become so general not only in 
the system of Leibnitz, but also in other systems, that it seems to have become fixed, 
without much regard for their original derivation and meaning ; and hence a change 
in that respect would appear like an unnecessary innovation. 

The term “ fluxion’’ is more truly significant of the true principles of the science 
then the term “‘ differential,’’ and the symbol ‘‘ d’’ is no better than some other would 
be ; but it is just as good as any, and the use of it involves no inconsistency with the 
principles on which the system is based. 


CONSTANTS. 


(4) The other class of quantities which enter into the 
transcendental analysis is that of constants. These are sup- 
posed to have a fixed value, although it 1s not always neces- 
sary that this value should be known or given. In equa- 
tions the constant quantities express the conditions of the 
proposition, and while they are generally supposed to have 
a given, or, at least, an asszgnable value, there are many Cases 
in which their value must be determined by the solution of 
an equation just as any unknown quantity in algebra is deter- 
mined. This, however, does not make them variables ; their 
value is as much fixed as if it were known at first. The 
solution of an equation is rendered necessary in order to 
make the immediate conditions conform to some ulterior 
conditions imposed upon them. Thus the general equa- 
tions of two circles will determine the curves when the con- 


DEFINITIONS AND FIRST PRINCIPLES. 53 


stants are given; but if there is an ulterior condition that 
they must be tangent to each other, the constants must be 
made to conform to this condition; which can only be done 
by an equation from which the necessary values can be 
obtained. This solution does not fx the values of the con- 
stants, but only makes known those values which were fixed 
or rendered certain by the conditions to which they were 
subjected. A constant then ts never in a state of variation. 


FUNCTIONS. 


(5) A function of a variable is any algebraic expression whose 

value depends on that of the variable. Thus 

GRE 32% 
is a function of x since its value changes with that of x, sup- 
posing a to be constant. The expression 

ax+ by 
is a function of x and y (a and 4 being constant), for it de- 
pends on both x and y for its value; and thus we may have 
a function of any number of variables. 

When the expression does not involve an egwation, the 
variables are independent of each other; that is, we may 
assign to any of them any value whatever without regard to 
the values assigned to the rest. But an egwation which con- 
tains variables will have at least oze dependent on the others 
for its value. Thus in the equation 

Wee eax 
in which x, y and w are variables, we may give arbitrary 
values to any two of them, but the value of the third must 
be determined from the equation. ‘This last 1s called a func- 
tion of the others and the dependent variable, while the oth- 
ers are called zudependent variables. When the dependent 
variable stands alone in one member of the equation it 1s 
called an expiicit function of the others, but when combined 


54 DIFFERENTIAL CALCULUS. 


with the others it is called an zmplicit function. The term 
function, however, applied to the dependent variable is to 
be understood as meaning the representative of the function, 
and not, literally, the function itself. 

Functions are commonly divided into two classes, which 
are distinguished by the manner in which the variables enter 
into them, and are called “A/eebraic ” and “ Transcendental.” 

Algebraic functions are those inwhich the variables are sub- 
jected only to the operations of addition, subtraction, multiplica- 
tion, division, and tnvolution or evolution, denoted by constant 
exponents or indices. 

Transcendental functions are those in which the vartable ts 
either an exponent, logarithm or trigonometrical line, such as a 
sine, tangent, etc. ‘This distinction 1s not important, however, 
and may be disregarded. 

(6) The fundamental problem of the calculus is to find 
the differential, or rate of change, in a function of a varia- 
ble produced by that of the variable itself 

As the differential or rate of change in a variable is rep- 
resented by a suppositive change, taking place at a uniform 
rate, and that of the function arising from it by a correspond- 
ing suppositive change, these changes (being uniform from 
the beginning)will have a constant ratio independent of their 
value. Hence the differential of the variable ts always a factor 
of the differential of tts function. ‘The other factor, that is, 
the ratio between the differential of the variable and that of 
its function, is called the differential coefficient of the function. 
Since this ratio is not affected by the value of the suppositive 
change representing the differential of the variable, this dif- 
ferential is indicated by an indeterminate symbol, and the 
differential of the function becomes a function of that symbol. 

The differential of a function is obtained by a process 
called afferentiation, and the differential coefficient is 
obtained by dividing the differential of the function by that 


DEFINITIONS AND FIRST PRINCIPLES. 55 


of the variable; so that the differential coefficient of ax®—dx 
would be 

aax*— hx) 

ax 
If we represent the function by wz we have 

U—= ax" —bx 
and , 

differential coefficient, which 
we can obtain as soon as we know how to find the differen- 
tial of ax?—dx. 

(7) If we have an equation containing variables in each 
member, since the two members are a/ways equal, their rates 
of change are also equal: hence we may differentiate 
each member as a separate function, and place the results 
equal to each other. If the equation contains more than 
one variable, one of them will be dependent and the value 
of its differential will depend on the values of the other 
variables and their differentials. Either of the variables 
may be taken as the dependent one, and it will then repre- 
sent a function of the rest. 

If the differential of a function of two or more variables 
be taken with reference to ove only, and then divided by z¢s 
differential, the result will be the differential coefficient for 
that variable, and all the rest must be treated as constants 
for that coefficient ; and the function as a function of ¢hat 
variable ; for a differential coefficient can exist only between a 
single vartable and its function. : 

If we wish to zuzdicate a function of any variable, as x, 
without giving it any particular form, for the purpose of 
demonstrating some general truth applicable to all forms, 
we use the expression F (x), which means any function 
depending on x for its value. If it is a function of two or 
more variables, the expression is F (x, y), or F (x, y, 2), and 
similarly for a greater number of variables, 


oF CAO Nee 


DIFFERENTIATION OF FUNCTIONS. 


PROPOSITION I. 


(8) Zo jind the sign with which the differential of the varia~ 
ble must enter that of the function to which tt belongs. 

Among the characteristics of quantity as used in the cal- 
culus, is that of being essentially positive or negative, ac- 
cording as it lies on one side or the other of the zero point. 
If, for instance, the value is reckoned toward the negative 
side it is essentially negative independent of the algebraic 
sign that may be prefixed to it. Thus the co-sine of 120° is 
essentially negative, whether it is to be added or subtracted, 
while the sine of the same angle is essentially positive. 
These characteristics are independent of, and wholly dis- 
tinct from, the algebraic signs that may be prefixed to them. 
Hence when quantities enter into a function as variables we 
must inquire what will be the effect of these characteristics 
on the influence which their rate of change will have on 
that of their function, so that we may give to the differen- 
tial of the variable that sign which will produce a rate of 
change in the function in the right direction. 

If, then, a variable, whether intrinsically positive or neg- 
ative, is increasing, z¢s raze of change will affect the rate of 
its function in the same direction as its value affects the value 

56 


DIFFERENTIATION OF FUNCTIONS. 57 


of the function; and, hence, having essentially the same 
character (Art. 3) it must have the same sign. 

If it is diminishing, its rate of change will affect that of 
its function in a direction conérary to that in which the value 
of the variable affects the value of its function; but having 
itself a character contrary to that of the variable (Art. 3) it 
must still have the same sign. 

Let us for instance take the function 

ne Be 

in which x is a positive and increasing variable. Nowif y 
is increasing its rate will be essentially positive or negative, 
according as y is itself essentially positive or negative (Art. 
3), and must therefore affect the differential or raze of the 
function (to increase or diminish it) in the same direction 
as y affects its va/we. If y is diminishing, its rate of change 
will be essentially negative if y is positive, and positive if y 
is negative (Art. 3), and will, therefore, affect the raze of the 
function in a direction contrary to that in which y affects its 
value ; but being essentially contrary in its nature to that of 
y, it must enter the differential of the function with the 
same sign in order to produce a contrary effect. 

Thus, whether the variable be intrinsically positive or 
negative, whether it is increasing or diminishing, “e sign 
prefixed to its differential must in all cases be the same as that 
prefixed to the variable itself. 


ILLUSTRATION. 


If we consider a northern latitude positive and a south- 
ern one negative, and there are two vessels, A and B, sail- 
ing north of the equator, let the latitude of A be represented 
by x and that of *B by y. Then the difference of their lat- 
itudes will be x—y. If both are sailing north their rates of 
progress wiJl be positive, and the rate of change in their 


58 DIFFERENTIAL CALCULUS. 


difference of latitude will be the difference of their rates of 
sailing; hence the rate of B, which is the differential of y, 
will have a minus sign. If B is sailing south the difference 
of their latitudes will be still x—y, but since. the rate of 
change in this difference is the real sum of their rates of 
sailing, the rate of B being essentially negative (Art. 3) 
must have a minus sign, so that the algebraic difference will 
produce the real sum. 

If B be south of the equator, the difference of their lati- 
tudes will be their real sum, but y being now essentially 
negative must have a minus sign to produce this sum, and 
it will still be expressed by x—y. If B be sailing south (A 
being still sailing north), the rate of B will be negative 
(Art. 3), and since the rate of change in the difference of 
their latitudes is-the real sum of their rates of sailing, the 
rate of B must have a minus sign, so that the algebraic dif- 
ference will be the real sum. If B be sailing north its rate 
will be positive (Art. 3), and since in this case the rate of 
change in the difference of latitudes will be the real differ- 
ence of their rates of sailing, the rate of B must still have a 
minus sign, so that the algebraic difference will correspond 
to the real difference. Hence, if y enter the function with. 
a minus sign, its differential or rate of change will have a 
minus sign, whether y is intrinsically positive or negative, 
or is increasing or diminishing. A similar result would fol- 
low if the sign were flus; the sign of the differential would 
be plus. 


Proposition II. 


(9) Zo find the differential of a function consisting of terms 
connected together. by the signs plus and minus. 

That is to say, to find the rate of change in the function 
arising from the rates of the variables which enter into it. 

Every term in an algebraic expression may be considered 


DIFFERENTIATION OF FUNCTIONS. 59 


as having a s¢ngle value, made up, of course, of the respec- 
tive values of the quantities that compose it, and their rela- 
tions to each other, and may, therefore, be expressed by a 
single letter. If a term contain none but constant quanti- 
ties the letter representing it will be considered as a con- 
stant. If it contain variables, the letter representing it will 
be considered as a variable having the same rate of change 
as would arise in the term itself from the rates of change in 
the variables which enter into it. 

It will, therefore, be sufficient to investigate the case of a 
function in which each one of the terms is represented by a 
single letter. 

Let us suppose some of thece terms to be variable and 
others constant, and the variables to be changing their values 
at any rate whatever, either uniform or variable, and each 
one independent of the rest. The constants will, of course, 
have no rate of change, and will, therefore, not affect the 
rate of change in the function. 

The differential of each variable will be the suppositive 
uniform change that qwould take place in it in a unit of time at 
the rate existing at the moment of differentiation; and the 
differential of the function is the uniform change that would 
take place in it arising from the supposed uniform changes | 
in the variables. Now, if we suppose this symbolic or sup- 
positive change to be made in each variable, the correspond- 
ing change in the function will be the algebraic sum of the 
changes in the variables; and as these are by supposition 
uniform, the change in the function will be uniform also at 
the rate at which it commenced, and will, therefore, be the 
symbol of that rate or the differential of the function. 

For example, let us take the function 

eee een ee at TY 
in which x, y and z are variables, and a, J and ¢ are con- 
stants, and represent by dx, dy and dz the differentials or 


60 DIFFERENTIAL CALCULUS. 


uniform changes that would take place in x, y and zina 

unit of time from the moment of differentiation. Let us 

also suppose these symbolic changes to take place; the func- 

tion would then become | 
x+ax—y—dy + a—b+ 2+d2+¢ 

(Art. 8) and if from this we subtract the primitive function 


we have 
dk—1y ae 


which represents the uniform change in the function aris- 
ing during the same unit of time from the suppositive 
changes in the variables. It is, therefore, the symbol repre- 
senting the corresponding rate of change, or differential of 
the function. Hence 
a(x—y+a—b+2+¢)=dx—dy+dz 

or the differential of a function composed of terms contatning 
zndependent variables, having any rates of change whatever, the 
terms being connected together by the signs plus and minus, ts the 
algebraic sum of the differentials of the terms taken separately 
with the same signs. 

Since each of the terms in the case given may represent 
a compound term of any form whatever, it is now necessary 
to examine the method of finding the differential of a single 
term in every form in which it may occur. 

The number of these forms for algebraic ternis is limited 
to seven as follows: 

rt. A variable multiphed by a constant. 
One variable multiphed by another. 
A variable divided by a constant. 
A constant divided by a variable. 
One variable divided by another. 
A power of a variable. 

7. A root of a variable. 

These simple forms, or some combination of them, which 
can be dissected and operated by the same rules, constitute 
all that can be assumed by single algebraic terms, 


ARO 


DIFFERENTIATION OF FUNCTIONS, 61 


(10) Zo find the differential of a variable multiplied by a con- 
stant quantity. 

We have seen that 

a (atyte+u)=dx+dy+ds+du 
If we make these variables each equal to x we shall have 
xty+s+u=4x 
and 
ax+dy+d2+du=4 ax 

hence 

d (4x)=4 dx 
As the same reasoning will extend to any number of terms, 
we may make the equation general, and we have 

d (nx)=n dx 
That is, the rate of change of wz times x is equal to z times 
the rate of change of x. 

Hence, the differential of a variable, with a constant coeffi- 
cient, ts equal to the differential of the variable multiplied by the 
cocficient. In other words, the coefficient of the variable 
will also be the coefficient of its differential. 


What is the differential of 4?4-+-c*z? — Ans. 
. What is the differential of a*dx+c?dy?— Ans. 


EXAMPLES. 

Ex. 1. What is the differential of abz?— Ans. abdsz. 
Hx. 2. What is the differential of d¢y?— Ans. b?dy. 
fix. 3. What is the differential of ax+cy ? — Ans. adx-+cdy. 
Ex. 4. What is the differential of x«—dy? — Ans. 
Ex. 5. What is the differential of (a+) «?— Ans. 
Ex. 6. What is the differential of (c—d) y? — Aus. 
Ex, 7. What is the differential of ax+dy-+cz? — Ans. 

8. 

9 


Ex. 10. What is the differential of a®y—J6?x ? — Ans. 
Ex. 11. What is the differential of 4(ay—cx) ? — Ans. 
Ex. 12. What is the differential of c?(dx-+az) ? — Ans. 


62 DIFFERENTIAL CALCULUS 


LEMMA, 


(il) If two variables are increasing at a uniform rate, their 
rectangle will be increasing at an accelerated rate, but the 
acceleration will be constant. 

Let x and y be increasing uniformly at rates represented 
by dv and dy. Then dx and dy will be the actual increments 
of x and yin a unit of time, and at the end of m such units 
ay will have become 

(x+mdx)( y+mdy)=xy+mydx+ mxdy +m? &xidy (1) 

In one more unit of time we shall have 

(xt(m+i)dx) (yt(mtr)dy)= xy t(m+1)ydet+(mt+r1)xdy 
+(m+1)?dxdy (2) 

In still another unit of time we shall have 

(x+(m+2)dx) (y+(m+2)dy)=xy+ (m+ 2) yde+(m+2)xdy 
+(m+2)*dxdy (3) 

Subtracting the second member of (1) from that of (2) we 

have 


pdx txdy +(2m+1)dxdy (4) 
and subtracting the second member of (2) from that of (3) 
we have 
ydx +xdy +(2m+ 3)dxdy (5) 
and subtracting (4) from (5) we have 
2dxdy (6) 


Now the expression (4) is the increment of the product 
arising from the uniform increments of the variable factors 
during one unit of time, and expression (5) is the increment 
of the product during the next equal unit of time arising from 
the next equal uniform increments of the variables. These 
increments of the product may therefore be taken to represent 
its successive mean rates of increase arising from the uniform 
increase of the variable factors during two equal successive 
units of time; and the expression (6), which is the differ- 
ence between these rates will represent their acceleration, 


DIFFERENTIATION OF FUNCTIONS 63 


Now since this last is a constant quantity and independent 
of m, it follows that the acceleration of the mean rates of 
increase of the product will be constantly the same during 
every two consecutive units of time, while the factors are 
increasing at a uniform rate. And since the increase of the 
variables is continuous and uniform, that of the product will 
also be continuous and according to a uniform law of some 
kind ; and since for every possible variation in the number 
and value of the units of time, and in the value of the rates 
dx and ay the acceleration of the rate of increase of the 
product is constant for successive periods, it must be so 
continuously, and equal to twice the product of the rates of 
increase of the variable factors. 


PROPOSITION IV. 


(12) Zo find the differential of the product of two independent 
variables. 

Let us suppose the two variables A and B to be increasing 
at any rate whatever, either uniform or variable, and inde- 
pendent of each other. Suppose also that when A has 
become equal to x, B will have become equal to y, and that 
dx and dy represent their respective rates of increase at that 
instant; then they will represent the uniform increments, 
that would be made by A and B respectively, in the same unit 
of time, at these rates; and these suppositive increments 
are what we have to consider. Suppose again that one-half 
of each increment be made immediately before A and B 
become equal to x and y, and the other half afterwards. In 
the first case the product of A and B, or AB, at the begin- 
ning of the increment will be equal to 

(x—34dx)(y— 4d) =ay— VYoydx— Vendy tif drdy 
and at the end of the unit of time it will be equal to 
(a+ %dx)( y+ Yay) =ayt Yydet Yxdyt Y dxdy 
Subtracting the first product from the last we have 
ydx+xdy 


64 DIFFERENTIAL CALCULUS. 


which represents the difference between the two states of 
the rectangle AB, or the increment made by it, while the 
factors are passing from x—'W%dx and y—Way to x+%dx 
and y+%dy ; that is, while the variables are receiving the 
uniform increments represented by @x and dy, their respec- 
tive rates of increase at the instant they are equal to x and 
y, the rectangle is receiving an increment represented by 
ydx+xdy. We are now to show that this increment repre- 
sents the rate of increase of AB at the moment that dx and 
dy represent the rates of increase of A and B separately, 
namely, at the instant they become equal to x and y. 

This suppositive increment would not, of course, be made 
at a uniform rate, but as we have seen (lemma) at a w- 
Jormly increasing rate. Wence when AB would become xy, 
and the variables had received half their suppositive incre- 
ments, the increment of AB would have received half the 
increase of its rate, which would then have become equal to its 
-mean rate for that unit of time. But the mean rate is that 
by which the increment zvou/d be made in the same time if 
it were uniform, and if the increment were made at a uni- 
form rate it would measure the rate of increase of the rec- 
tangle. Now the actual increment (represented by ydx+.xdy) 
of the rectangle, being made in the same time, as it would 
be if made uniformly at its mean rate, existing when A and 
B were equal to xand y, zs the true measure of that rate ; that 
is, of the rate of increase of AB at that instant; or of xy if 
we consider x and y as the variables. Hence “he differential 
of the product of two variables ts equal to the sum of the products 
arising from the multiplication of each variable by the differen- 
ential of the other. 

Notr.— This proposition being the ey to the whole subject of the differential cal- 
culus, should be carefully studied and well understood. The result of this proposition 
might have been surmised by considering that a product of two variables is subject to 


two independent causes which produce its rate of change. If x has a certain rate of 
increase, that of the product will, from ¢Za¢ cause be y times that rate; andif y havea 


DIFFERENTIATION OF FUNCTIONS. 65 


certain rate of increase, that of the product will from ¢a¢ cause be x times that rate: 
and the total effect of both causes will be the sum of the partial effects arising from 
each cause independent of the other; that is, the entire rate of-increase of xy, is y 
times that of x plus x times that of y. This, however, is not #athematical proof — 
it only makes the result Arodadle, Yet the text books assume without any better proof, 
that the total differential is equal to the sum of the partial differentials, which is equiv- 
alent to the proposition as I have stated it. The truth 1s,it cazzot be proven math- 
ematically by the method of infinitesimals nor limits. 


This proposition may be illustrated geometrically thus: 
Let x be represented by the line 4B (Fig. 1), and y by 
the line AC; then the product «xy will be represented by 


the rectangle ABDC. Suppose , nes te: 
x and y to be each increasing in ‘ | 
such a manner that when x has © E 


become equal to AB, and is 

then increasing at a rate, that, 

if continued, would produce the 

increment #4’ in a unit of time, 

y will have become equal to 4 Neary 
AC and be increasing at a rate Fig 1. 

that would produce the increment CC’ in the same unit of 
time. Then BBA’ will be dx and CC’ will be a, for they 
will be the true symbols of the vats of increase of x and y. 
Now the uniform increment, B24’, of x will produce the 
uniform increment BD £# SA’ of the rectangle, which will 
therefore represent z/s rate of increase arising from that of 
x. In like manner CC’ # D will represent the rate of in- 
crease arising from that of y. Hence the symbol of the 
entire rate of increase of the rectangle arising from the 
rates of increase of the sides, will be the sum of the two 
rectangles BB’ EL D and CC’F D or ydx+xdy. 

It must be remembered that the increments DE and DF 
are not actual increments given to the sides BD and CD, but 
suppositive increments, or symdols, representing their rates 
of increase; for in order that the suppositive increment of 
the rectangle may be uniform, the sides BD and CD must 

5 


66 DIFFERENTIAL CALCULUS. 


remain constant. Hence the two rectangles BB’ED and 
CC’FD are not actual increments of the rectangle ABDC, 
but suppositive increments, winca are symbols showing what 
would be its increment if the rate were to remain constant 
from that point in the value of x and y for a unit of time, 
and are, therefore, the measure of that rate. 


Note.— The increment of the rectangle arising from the actzaZ increments of x 
and y would include the small rectangle DED’F or dx.dy; and hence it would be too 
great to represent the required rate by so much; and it is to get rid of this surplus that 
the advocates of the infinitessimal theory, who take the actual increment to represent 
the rate, reduce it to an infinitessimal, which they claim may be neglected with impu- 
nity. But we see the true reason for throwing it out of the expression for the rate is, 
not because of its insignificance, but because, being produced by the actual increments 
of + and y after they have passed Jevond the values of AB and AC, it has no connec- 
tion with the rate at which the rectangle was increasing at the moment x and y were 
egual to those values. 


PROPOSITION V. 


(13) Zo jind the differential of the product of any number 
of variable factors. 

Let us first take the product of three variables as 

XYZ 
If we represent the product xy by z we shall have 
xyes 
and (Art. 7 and 12) | 
a (xyz) =a(uz) =sdu+udz 
Replacing w by its value we have 
A xys)=2d(xy)+xydz 


Uxy)=yde+xdy 


or since 


we shall have 


A xyz) =sydx texdy +xyds 
If we take the product of four variables as 


and make xy=r and zw=s we have 
Ty2u — 7s 


A xysu)=a(rs)=rds+sdr 


and 


DIFFERENTIATION OF FUNCTIONS. 67 


Replacing 7 and s by their values we have 
a xysu)=xyd( su) +2uad(xy) 
substituting the values of azz) and d(xy) we have 
Uxy2u)=xysdut+ayuds+usxdy +usydx 

These examples may be carried to any extent, but are 
enough to show the daw which governs the differential of a 
product, which law may be thus stated: The rate of change 
in a product arising from the rate of any one factor is equal 
to the product of all the other factors multiplied by that 
rate; and the total rate of change in the product is the sum 
of all the partial rates arising from the rates of the factors 
taken separately. In other words, the total effect arising 
from all the causes acting together is equal to the sum of 
the partial effects arising from each cause acting separately. 

Hence, the aiffcrential of the product of any number of vart- 
able factors ts equal to the sum of the products arising from mul- 
tiplying the differential of each variable by the product of all the 
other varicbles. 


EXAMPLES. 


Ex. 1. What is the differential of ayz? — 
. Ans. aydz+azdy 
“x, 2. What is the differential of 4 dry? — 
Ans. 4 0(xdy+ydx) 
Zx. 3. What is the differential of ax+édyz? — 
Ans. adx+byds+bedy 


£x. 4. What is the differential of xy—wz?— Ans. 

£x. 5. What is the differential of 3ax—2xy ? — Ans. 
Ex. 6. What is the differential of 2ay-+3?— Ans. 

Lx. 7. What is the differential of 4abxyz? — Ans. 

Lx. 8. What is the differential of dcw—a? —zy ?— Ans. 
Lx. 9. What is the differential of 4axy—b? +cu? — Ans. 


£x. 10. What is the differential of 7a? —2bu-+cy? — Ans. 


68 DIFFERENTIAL CALCULUS. 


PROPOSITION VI. 


(14) Zo find the differential of a fraction. 


CASE I. 
Let the fraction be 
x 
n 
in which the variable is divided by a constant. Make 
7 ee 
Picts 
then 
x=ny 
and (Art. ro) 
ax=ndy 
hence 
Nene 
yaa =e 
that is, 


The differential of a fraction having a variable numerator 
and a constant denominator 1s equal to the differential of the 
numerator divided by the denominator. 


CASE 2. 


Let the fraction be 
n 


x 

in which the denominator is variable and the numerator 
constant. Make 

n —_— 

re Pos 
then 

AV nH 
and (Art. 12 and Art. 9) 

axy)=ydx +xdy=dn=o 


hence 
yax 
x 


dy = 


DIFFERENTIATION OF FUNCTIONS. 69 


Replacing y by its value we have 


=a 
Hn)\= eer ge 
Goa og aN 
that is, 


Lhe differential of a constant divided by a variable ts equal to 
minus the numerator tnto the differential of the denominator, 
divided by the square of the denominator. 


CASE 3. 


Let the fraction be 
we. 
y 
in which both terms are variable. Make 


Male 
y 4 
then 
x=uy 
and 
adx=duy)=ydu+udy 
hence 


Ape __ dx—udy 


Replacing z by its value we have 


Ae jee =a _ ydx—xaly 


y 


that is, 

The differential of a fraction of which both terms are vart- 
able, ts equal to the differential of the numerator multiplied by 
the denominator, minus the differential of the denominator mul- 
tiplied by the numerator, and this difference divided by the square 
‘of the denominator. 

If the variables in any of these cases should be functions 
of other variables, we can represent them by single letters 
and replace them in the formula by their values. 

Thus, suppose the given fraction to be 


70 


make 


then 


DIFFERENTIAL CALCULUS. 


eat 
“y 


u—x=s and vy=r 


u—x ras—sdr 
Ast) = Kt) ee 


Replacing the values of s and ~ we have 


Le. vydu—x)—(u—x)d(vy) 
he el) = ra 


which being expanded becomes 
ez a! vydu—ovydx—uvdy—uydy +oxdy +xydy 


Yale PS a ie 
LUN eae s 
a mes 
L58; A. 
Lh; 3, 
Tie ea oy 
oe! 
ToS: 
La 9: 
ESE LO 
Lane 
L212 


vey? 
EXAMPLES. 
What is the differential of ax+5 ? 
4 
Ans. adx + 
What is the differential of nec 


Ans. ie 


What is the differential of a Ans. 
What is the differential of Se Ans, 
What is the differential of sxe? Ans. 


What is the differential of 3(a—x)—2-4? Ans. 
What is the differential of zax+x(y—c)? Ans. 
What is the differential of er x)\(u—c)? Ans. 
What 1 is the differential of 5 —— x"? Ans. 
What is the differential of ao Pest z)? Ans, 
What is the differential of 6ab—3xy(u—c)? Ans. 


What is the differential of (s—y)}(a—-) ea =? Ans. 


DIFFERENTIATION OF FUNCTIONS. 71 


Proposition VII. 


(15) Zo find the differential of any power of a variable. 

We have seen (Art. 13) that the differential of the product 
of any number of variables is equal to the sum of the pro- 
ducts arising from multiplying the differential of each vari- 
able by the product of the rest. If there are z variable fac- 
tors in the given product, there will be z products in the 
differential, and the coefficient of the differential of each 
variable will be the product of (z—1) variables. If now all 
. the variables become equal, we may represent each one by 
x, and the product will become x”, while each of the pro- 
ducts composing the differential will become x”—1¢dx, and 
as there are z of these products, the sum of them will be 
nx"—17x, hence we have 

a(x?) x 

Hence, the differential of a variable raised to a power ts equat 
to the variable raised to the same power less one, and multiplied 
by the exponent of the power and the differential of the variable 

If, for example, we take 

d (xyzu) =xyzdu+xyuds+xzudy +ysudx 
and suppose all the variables to become equal, we shall have 
axt=4x% dx 
If we represent x” by y we shall have 
dy=nx" lax 
and 
Bet 
which is the a&fferential coefficient of that function of x. 

(16) The rule here given, for finding the differential of 
the power of a variable, holds good for all values of x, 
whether integral or fractional, positive or negative: 

Case 1. Let 2 be negative and represented by —™m, then 


72 DIFFERENTIAL CALCULUS. 


d (Art. 
ey nes Be Gore ete 
(zm) = ae 


am x2 AS xm 


Dividing both terms of this fraction by «”~1! we have 
max 


— 


 ymt1 


—mx-™17x 


in accordance with the rule. 
Case 2, Let 2 be equal to + then 
1 
xm =m 
i 
Representing x” by y we have 


xy 
and 
ax=my"—ldy 
or 
ax 
Dyna 


Replacing y by its value we have 


Ce is SCE 8 2, 


mee 7 1 wm 
M\ xm mx a, 


Case 3. Let 2 be equal to = and represent «” by y and 


we have 
T 


yorum or ayn aay ft 


and 
mylqy =rx"—lax 
whence 
yom rat —ldx 
a mymt 


Replacing y by its value, we have 
$3 


Ue r—1 
At = me ; =a" 1- mim Dele = sem ax 


m r\m—-t1 
(a) 


hence the rule is true in all cases. 


DIFFERENTIATION OF FUNCTIONS. 13 


Proposition VIII. 


(17) Zo find the differential of any root of a variable. 


Let us take the function 4/x and make it equal to y, 
then 


mes 
and 
my"-lgy =ax 
whence 
_ ae 
= myn 
Replacing y by its value we have 
eae ax 
ay/x — mn ~m— m—1 


That is, the differential of any root of a variable ts equal to 
the differential of the variable divided by the index of the root 
into the same root of the variable raised to a power denoted by 
the index of the root less one. . 

Hence 


Ef x= fos 


These results could of course be obtained under the pre- 
ceding rule, by giving to the variable a fractional exponent. 
But this is not always convenient. 

(18) In all the cases which have been explained under 
these eight propositions, the single letters that have been 
used may represent functions of other variables, in which 
case the operation must be continued by substituting the 
functions for the letters representing them and performing 
upon ¢#em the operations indicated, which can be done by 
the rules already given; for the terms of all algebraic func- 
tions must ultimately take some one of the forms that we 
have discussed. 


74 DIFFERENTIAL CALCULUS. 


Thus, if we have the function 
ee ieW 
a—x 
we may represent the numerator by w and the denominator 
by v we shall then have 


a—x v 
and 
x*—A/y vau—udv 
Sas RD I eT ET 
a—x v 


Replacing w and v by their values we have 
aay (a—2)d =f) (a? — of alas) 
a—X% (a—x)? 


and performing the operations indicated we have 


= dy ~ 
ay (a—x)(20ux—3" =) +(x? —4/ y)dx 
a—x (a—x)? 
which completes the differentiation. 


PROPOSITION IX, 


(19) To find the differential of a function with respect to 
the independent variable when it enters the given function 
by means of another function of itself. 

Let there be a function of y in which y represents a func- 
tion of x. The proposition is to find the differential coeffi- 
cient of the given function with respect to x. Represent- 
ing the given function by w we have 

u—F\(y) and y=F(2) 

In order to find the relation between dz and dx, it might 
be supposed necessary to eliminate y from the equations and 
obtain one directly between zw and «x, to which the ordinary 
process of differentiation could be applied, But this is not 


DIFFERENTIATION OF FUNCTIONS, 75 


necessary; for if we differentiate these equations as they 
are We have 

du=F'(y)dy and dy=F"'(x)dx 
in which #"(y) and #’(x) represent the differential coeffi- 
cient of ~ with respect to y and of y with respect to x. 


From the first of these we obtain 
du 


LW =F 
and placing this value of ae to the other we have 
FG} =F'"(x)dx or — = Fy). Zita) 

That is, the differential seen of wz with respect to x 
is equal to that of z with respect to y multiplied by that of y 
with respect to x. Hence 

When the variable of a given function represents the 
function of an independent variable, then “he aifferential 
cocfficient with respect to the independent variable ts equal to the 
product of the differential coefficient of the function with respect 
to the given variable, multiplied by the differential coefficient of 
the given variable with respect to the endependent variable, 

Thus if we have the function y?—ay in which y=2a—x?, 


representing the given function BE wz, we have 
a 


=2y— —a and ° 2k 
whence 


gu Set Ginx 
We 20X — 4xy =4x3 —6ax 


EXAMPLES. 


Find the differentials of the following functions in which 
the variables are independent. 


at, axe Ans. 2a°xdx-+-az 
eX. 2, 0x? —y>--a Ans. 2bxdx—3y* dy 
ae 3. ax” —bx? +x Ans. 

Ex. 4: (¢c+d)(y?—<x*) Ans, 


mato: 
Joly 
apes 
seo ee Ys 
as Oi 


Se 
raLOe 


axe iy 


LS: 


LQ: 
Pezey, 
A eAG 
fae 
ey 
ee 
4 Pas 
220; 
73 
ee Os 


Oo or Qe 


DIFFERENTIAL CALCULUS. 


54° —2ay—b? 
xr — x3 +46 
ax — 3bx 
(x? +a)(x—a) 
wry? — 32 
ax*(x> +a) 

a 
b—2y* 
Ne me 
NV oax+x® 


2 


VJ — x? 
Be 

xt 1— x? 
(aaa) E 

a2 —x2 

ah 

1 +x? 
Vira— Vi-« 
Ae eA lr 


(a + bcnym 


Ans. 
Ans. 
Ans: 
Ans. 
Ans, 
Ans, 


Ans. 


Ans. 
Ans. 
Ans, 


Ans. 
Ans. 
Ans. 


Ans. 


Ans, 


Ans. 


Ans. 
Ans. 


Ans. 
Ans, 
Ans. 
Ans. 


Ans. 


Ans. 


DIFFERENTIATION OF FUNCTIONS. v9: 


Ex. 20; (6—c)(x—y)® Ans. 
02-80. AV 22 +a n/x Ans. 
£x. 31 <A person is walking towards the foot of a tower, 
on a horizontal plain, at the rate of 5 miles an hour; at what 
rate is he approaching the top, which is 60 feet high, when 
he is 80 feet from the bottom? 

Let the height of the tower be a, the distance of the per- 
son from the foot of it be x, and the distance from the top 
be y; then 


yr=atx? 
and , 
ydy=xax 
whence 


a= ade _ 80 ft. x 5m. __ 


Ans. 4 miles per hour. 

Ex. 32. Two ships start from the same point and sail, 

one north at the rate of 6 miles per hour, and the other east 

at the rate of 8 miles per hour; at what rate are the ships 
leaving each other at the end of two hours ? 

Ans. 10 miles per hour. 


Ex. 33. Two vessels sail directly south from two points 
on the equator 4o miles apart, one sails at the rate of 5 
miles per hour, and the other at the rate of 1o miles per 
hour; how far will they be apart at the end of 6 hours, and 
at what rate will they be separating from each other suppos- 
ing the meridians to be parallel? Ans. 3 miles per hour. 

Lx. 34. A ship sails directly south at the rate of ro 
miles per hour, and another ship sailing due west crosses her 
track two hours after she has passed the point of crossing, at 
the rate of 8 miles per hour; at what rate are they leaving 
each other one hour afterwards? 

Ans. 11.74 miles per hour nearly, 


£x. 35. The vessels sailing as in the last case, how will 


78 DIFFERENTIAL CALCULUS. 


the distance between them be changing, and at what rate, 
one hour before the second crosses the track of the first? 
At that time the first vessel will be 10 miles from the point 
of crossing, and the second 8 miles. Calling the distance 
of the first x, and the second y, the distance between them ° 


is V x? +y? which we will call w, then 
u=/ x+y? 
and 
AIR VEY) OO 04. 
OV a2 fp y® VY 100-+64 | 

We make 64 in the numerator negative because y is a posi- 
tive decreasing function, and its differential is therefore 
intrinsically negative. : 

From the above equation we find that the vessels are sep- 
arating at the rate of 1.56 miles per hour nearly. 

£x, 36. The height of an equilateral triangle is 24 inches, 
and is increasing at the rate of two inches per day; how 
fast is the area of the triangle increasing ? 

Ans. 32/3 square inches. 

Lx. 37. The diameter of a cylinder is 2 feet, and is in- 
creasing at the rate of 1 inch per day, while the height is 
4 feet and decreasing at the rate of 2 inches per day; how 
is the volume changing? and how the convex surface ? 

Ans. The volume is increasing at the rate of 288 = cubic 
inches per day. The area of the convex surface is not 
_ changing , 


5 Cab LO No DLT. 


SUCCESSIVE: DIFFERENTIALS. 


(20) In considering the differential of a function hitherto, 
we have regarded it as immaterial whether that of the inde- 
pendent variable was itself variable or uniform. In consid- 
ering the rate of change in the @ferentzal of the function, 
it will be most convenient to consider that of the indepen- 
dent variable as constant ; and this we have a right to do, 
since, the variable being independent, its rate is always 
assignable. 

If we have the product of two or more independent vari- 
ables, whether they are alike as x, or different as x.y.z, the 
value of the rate of change depends not only on that of each 
independent variable, but also on the absolute value of all 
the variables. Thus the value of a(x), or 3x*@x, depends 
not only on that of gx, but also on the absolute value of x?, 
and is greater or less as x® is greateror less. So that 3x*dx, 
which is the rate of change of x3, has its own rate of change 
or differential. 

To find this second differential we must treat the first as 
an original function; and as x is supposed to change uni- 
formly, ¢ will be regarded as a constant. Now the func- 
tion 3%°¢x may take the form 3¢x.x*, and 

W2de.x%" \= 30x. a x*)\=3dx.2xdx=bxdx? 
This is called the second differential of «?, being the differ- 
79 


8o DIFFERENTIAL CALCULUS. 


ential of the differential. This order is indicated by plac- 
ing the figure 2 as a sort of exponent to the letter d, thus 
d?(x) is the symbol for the second differential of «3, and 
hence 

d*(x?)\=6xdx? 

If the function is at all complicated, and, especially, if 
we desire to indicate a differential coefficient, it is much more 
convenient to represent it by a single letter; in which case 
the letter itself is, for the sake of brevity, called the func- 
tion. 

So that if we represent «x? by zw we shall have 

u=—x® and a@u=bxdx* (1) 

Since the second member of this last equation still con- 
tains the letter x, it will still have a rate of change which 
may be found by considering 6¢%* as the constant coefficient 
of x, and differentiating we have (Art. 10) 

Lu=b6dx* dx=6ax8 (2) 

The expressions @x* and dx are to be understood as 
indicating, zof the differentials of «* and «3, but the square 
and cube of dx. 

The figure placed as an exponent to the letter Zindicates 
the order of the differential ; and the differential of any order 
above the first is the rate of change in the differential of the 
previous order. The differentiation of the differential can 
only take place while the latter represents what is still a 
function of the independent variable. The differential of 
an independent variable, being, as we have stated, supposed 
to be constant can have no differential. 

If we divide equations (1) and (2) respectively by dx? 
and 7x*, we have f 

Cus Cu 
in which the second members are the second and third dif 
ferential coefficient of the function «°, 


SUCCESSIVE DIFFERENTIALS. SI 


(21) To illustrate the principle of successive differentia- 
tion, let us suppose 4A BDCFEG (Fig. 2) to represent a 
cube, of which the side 4 # is an increasing variable repre- 
sented by x. Let the suppositive increments Dd, Dd’ and Di’ 
represent the three equal rates of increase of the three x’s 
whose product is the given cube; we are to find what will be 
the corresponding rate of increase of the cube itself. That 
is, Dd being equal to @x, what is the value of a(x). 

At the moment the sides of the cube become equal to «, 
the cube tends to expand by the movement of the three faces 
DA, DF and DG outward in the directions Dd, Da’ and 
Da", each face continuing parallel to itself, and thus increas- 
ing the cube. But in order that these increments may rep- 
resent the raze at which the cube was increasing when they 
began, they must be made wznzformly at the same rate. 
Hence the areas of the faces must remain constant, and they 
must move at the 
same rate as the in- 
crements Dd, Du’ 
and Dd" are des- 
cribed; the move- 
ment being = con- 
trolled by those in- 
crements. Thus the 
three soids Da, Df 8 ag 
and Dg, generated : 
by the flowing out 
of the surfaces DA, Gh-1-) g 
DF and DG, with 


an unchanging area, 


at the same rate as 8 Bei 
when their sides be- Fa 
a SS SS ee ere ata 
came equal to x, L 
Fig. 2. 


form the increment 
6 


82 DIFFERENTIAL CALCULUS. 


that would take place in the cube in a unit of time, at the rate 
at which it was increasing when its side, or edge, became 
equal to x; and hence they form the true symbolic incre- 
ment which represents the rate of increase or differential of 
x3, Now each of the solids thus formed is equal to «*dx, 
and hence 
Ax?) =34%dx 

But if the cube, when its edge is equal to x, is in a state 
of increase, the faces DA, DF and DG have other tenden- 
cies besides that of flowing directly outward. TZhey also tend 
to expand in the direction of their own sides, and this ten- 
dency is quite distinct from the other. Let us examine and 
measure it. The tendency of the face DF to expand, aris- 
ing from that of the cube itself, would be by the flowing out 
of the sides DC and DZ in the directions Dd@ and Dua’, and 
remaining parallel to themselves. The vaé of increase of 
the face D/ would be measured by the areas described by 
these flowing lines in a unit of time, at the same rate as 
when they became equal to x, their lengths being constant ; 
that is, by the areas Dc and De’. But the face DF of the 
cube is the base of the solid Df, and the tendency of the 
base to expand imparts a like tendency to the solid, and the 
rate at which the solid tends to expand is such that while 
the base would increase by the rectangles De and De’, the 
solid would increase by the solids De’ and De", which there- 
fore represent, symbolically, the rate of increase of D/. 
Now De’ and De" are each equal to x@x*, and hence the rate 
of increase of D/ is equal to 2xdx*. But the differential 
of the cube is represented by three such solids as Df and 
the rate of increase of the whole, or the differential of the 
differential x? is 6xdx?. 

Again the solid De’ or xx? tends to increase in the direc- 
tion Da’ at such a rate as would generate the suppositive 
increment Dd”, equal to dv? in the same unit of time and 


SUCCESSIVE DIFFERENTIALS. 83 


in a uniform manner. Hence a(xdx*) is equal to dx, and, 
therefore, 2(6xd¢x*) is equal to 6¢x3. 

(22) It must always be remembered that the solids rep- 
resented in the figure are zof the actual increments of the 
cube, but the symbols which represent its rade of increase and 
the successive rates of that rate at the instant that x is equal 
to 44; that is, they are the increments that would take place 
in the cube, and in the increments themselves if made uni- 
formly. The Zaw which governs the increase of the cube, 
contains within itself not only the rate at which the cube is, 
at any instant increasing, but also the law of change in that 
rate, and the law to which that law is subject; and these 
symbols represent the development of that law which was 
actually operating at the instant the cube attained the value 
of 4483 or x«%, and before any farther increase had taken 
place. 

(23) We may learn from this demonstration the method 
by which the actza/ increment of a power may be developed. 
By dissecting the figure (Fig. 2) and noticing the parts of 
which the increased cube is composed, we find, jrs¢, the 
original cube or x; sccond, the three solids Da, Df and Deg 
or 3x°d@x ; third, the three solids De’, De" and Dé which 


. eee a 
represent Aa/f the rate of increase of 3x*d¢x or oer and 


fourth, the solid or small cube Da’, which represents one- 


Gait? 62x32 
ODT as 3? and these 


third of the rate of increase of 


make up the volume of the cube after being increased by 
the addition arising from the increment @y to the side AZ. 
But these increments are suppositive, and are used merely 
as symbols to show the successive rates of increase, all of 
which exist in the function x? before any increment actually 
takes place. 

If we divide 3x*@x by dx we shall have the first differen- 


~ 


84 DIFFERENTIAL CALCULUS. 


6xdx? 
2 


tial coefficient of «*. If we divide by adx* we shall 


have fa/f the second differential coefficient of «3; and if 
Oi bee 

we divide — by @x? we shall have one-sixth of the third 
ee | 


differential coefficient of «%; and these results, viz.: 3x, 
“2 and oe 
powers of @v must be multiplied in order to make up the 
parts composing the suppositive increment of +?. Now 
since these partial increments taken together with the orig- 
inal cube form also a complete cube, if we make dx a real 
increment and multiply its successive powers by these same 
coefficients, we shall have an actual increment of the cube, 
and the original x? will have become (x+a@x)8. 

It must not be forgotten that these differential coefficients 
are true of the cube defore the increment takes place, and 
when @x is equal to zero. 

For convenience we will designate the variable cube by 
wz, and in order to mark the point where the differential is to 


are the coefficients by which the successive 


be taken we represent its variable edge by (4-+.x) in which 
hi represents the side 4, or that particular value of the 
variable where the differential is to be taken, while x will 
represent its variable increment and take the place of dx. 
Then the cube 4Z or 48° will be represented by z or 
(z-+-x)* at the time when the variable w equals 4 or x=o. 
This being premised we take the differential coefficients 
already found, namely, 3x?, ad and =) and substitute (2-++.r) 


for x (reducing x to zero at the same time) and «x for dx in 
the other factors; then 3x”? becomes the first differential 
: ° if ° 
coefficient of w or (2+.)3, that is {-=3(4+.2)? with x=, 
6( + wx) 
2 
second differential coefficient of w with x=o,; and = 


Orgies: > becomes with «=o, that is 3%, or half the 


SUCCESSIVE DIFFERENTIALS. 85 


becomes one-sixth of the third differential coefficient of z or 
Ou = 
6ax3 


Hence, indicating by a vinculum that x has been made 


a 


: 6x 
equal to zero, we have for the three coefficients 3x, 5 and 


6 : F 
3-3 Which were true of the cube before any increment was 


made, 
) 2, 3 
Chas ersas) 


3B, . 


also true of the cube at the same time and the different parts 
of the cube increased will be represented as follows : 
The cube 4Z by (wz) or 23. 


The three solids Da, Df and Dg by (S)x or 3h? x. 


9 
w 


: Zt 
The three solids De’, De" and Dé by (< Jac? OF 34%; 


2ax* 
aru 


ees Ped Ae 8 
a Ole 
Ory, ; 


The solid Da” by ( 


and these make up the value of (4+.)’, hence 


wor (its)? =(u)+(Fe) « +(SGs) a? +(=— a) 8 


or 
(u+x)3 =h8 + 3h? x+ 3hx? +23 
This illustrates to some extent the law which connects 
together the parts which go to make up the change in the 
function of a variable arising from that of the variable itself. 
A more complete and general demonstration of this law is 
contained in the following theorem. 


MACLAURIN’S THEOREM. 


(24) A function of a single variable may often be ex- 
panded into a series by the following method. 


86 DIFFERENTIAL CALCULUS. 


Representing the function by wz and the variable by x we 
shall have 

: “i Pia) 

When this function can be developed, the only quantities 
that can appearin the development, besides the powers of 
x, will be constant terms and constant coefficients of those 
powers. Hencethe developed function may be put into the 
following form: 

Uu=At+BoetCxr? +-Dx3 +Fxt+ etc. (1) 
in which 4A, #, C, D, Z, etc., are independent of x. The 
problem is to find the value of this constant term 4, and 
the values of the constant coefficients B, C, D, #, etc. For 
this purpose we differentiate equation (1) successively, and 
divide each result by dx, the successive differential coeffi- 
cients will then be 


ai 
we B+ 2Cut 3 Da? + 4Exe + ete. (2) 
au 
Tae =2C+2.3.Dxt3.4.Ex® + ete. (3) 
aru 
Tye =2-3-D42.3.4. Ext ete, (4) 


Since x is an independent variable, these equations are 
true for all values of x, and, of course, when x=o. 

Reducing x to zero in equation (1), 4 becomes equal to 
the original function with x reduced to zero. We will rep- 
resent that state of the function by (w),; and also indicate by 
brackets around the differential coefficients that «=o in their 
values also. Then from equations (2), (3), (4), etc., we have 


a=(=), c=(5*), 28 p=(“%) 


and so on to the end of the development, if it can be com- 
pleted; if not, then in an unlimited series. 

Substituting these values of 4A, B, C, D, etc., in equation 
(1) we have 


SUCCESSIVE DIFFERENTIALS. 87 


which is Maclaurin’s theorem. 


+ etc. 


2 


EXAMPLES, 


Ex.1. Expand (a+)” into a series. 
Represent (2a+.x)” by z and we shall have 

u=(atx)"” =A+ Bet+Cxr?+Dx3 + etc. (1) 
Differentiating we have 


Tu 
=nla+x)"-1 
VT 
ee le 1) (a+x)"-? 
aru 
beaks 
dtu 
err: =n(n—1)(2z—2)(n—3(a+x)"—4 


=n(n—1)(z—2)(atx)"-8 


from which, when «=o, we have 


A=a” 
Te ie oe 
re mn) -, 
Pe nin—1)(~—2) P 
On2 
hy re aN 5 ty \ 2 
Pu n(n—1)(nz—2)(2z—3)a 
ee ore 


Substituting these values in equation (1) we have 
(n —1) n—1) (n— 
u=(atx)” =a" +na"-1x Se Aa BE ft—3.8 
and so on; the same result as by the binomial theorem. 


ih. : 
x. 2. - Develop. <-, into'a series. 


We have by differentiation 


88 DIFFERENTIAL CALCULUS. 


MU I 
ax (a+x)® 
ee eee 
dx® ~(a+x)3 
au 2a 


dx® ~~ (a+x)4 
and so on. 
Making «=o in the values of w and the differential coef- 


ficients we have 


= (=e (G=2 (=F 


Substituting these values in Maclaurin’s formula we have 


Tory Ely in eee 


Sis poe to 
atx a Set qi 


1) ie 


Pie eg EVElOp ine LENCO ss ae : ; into a series. 


Ans, 1 pate t+yr*+txr?+xr4+4+ etc. 


Ex. 4. Develop the function ee into a series. 


Ans. pe gt ye 5 qt Py? 7 maa ee 
I ; | , 
wig Wevelop Gaye aseries. Ans. 
#ix.6. Develop 34/(a+-x)2 into a series. Ans. 
fine Develop A a? +x into a series. Ans. 
Xo LION EID We = into aseries. Ans. 


Ex. 9. Develop (a?—x?)~ Ohrid series: Ans. 

The formula of Maclaurin apples in general to all the 
functions of a single variable that are capable of successive 
differentiations. But there are cases in which the function 
or some of its differential coefficients become infinite when 
x=o ; in such cases the formula will not apply. The func- 


ho: =n, & : 
tion, c+ax* is an example of this kind; for if we represent 
it by « we have 


SUCCESSIVE DIFFERENTIALS. 89 


and 


If in this we make x=o for the value of the coefficient 

B, we have 
B=—=% 

In general, any function of x in which x is not connected 
with a constant term under the same exponent, cannot be 
developed by this theorem; for the differential coefficients 
will be such as to reduce to zero or infinity in every case, 
when x is made equal to zero. 


TAYLOR'S THEOREM. 


(25) Zhe object of this theorem ts to obtain a formula for the 
development of a function of the sum or difference of two vari-— 
ables. 

The principle on which this theorem is based is the fol- 
lowing: The differential coefficient of a function of the sum 
or difference of two variables, will be the same whether the 
function is differentiated with respect to one variable alone, or 
to the other variable alone. Thus the differential coefficient 
of (x+y)” will be w(x+y)"—-! if we differentiate with res- 
pect to either variable alone, the other being considered as 
constant. 

A function of the sum or difference of two variables is one 
in which both are subject to the same conditions, so that the 
value of their sum or difference might be expressed by a 
single variable without otherwise changing the form of the 
function; and hence we may regard this sum or difference 
as itself a single variable. Now any rate of change in one 
of the two component parts (the. other being regarded as 


go DIFFERENTIAL CALCULUS. 


constant) will produce the same rate in the compound vari- 
able (so to speak) as it has itself; thus x+y will increase at 
the same rate as x if y be constant, and at the same rate as 
y if x be constant. So that changing froin one to the other 
is merely changing the rate of the single variable that would 
represent the value of their sumordifference. But such 
change in the rate will not change the va#o which it bears 
to the corresponding rate of the function (Art. 6); that is it 
will not affect the differential coefficient. 

(26) To apply this principle let us take any function of 
the sum of two variables, as (x+y), which we will repre- 
sent by w. If it can be developed into a series, the terms 
of the series may always be arranged according to the power 
of y,; the coefficients being functions of x and the constants ; 
hence it may be made to take the following form : 


u=F(xty)=A+By+Cy?+Dyi+LZLy4+ etc. (1) 
im which 4, B,C, ), 2, etc, are, indéependentoiy sat 
functions of «x. 
If we differentiate equation (1) regarding y as constant, 
and divide by ax, we shall have 


Lu Ae OB ea. aD , 
ax SE RETO erg Dot tes 


If we regard x as constant, and divide by dy, we have 


V2 
ue +42y?+ etc. 


. du - 
and since =, 1s equal to the second members of these 


a 
equations are equal; and since this equality exists for every 
value of y, and since the coefficients are independent of that 
value, the corresponding terms containing the same powers 
of y must be equal each to each; hence 


SUCCESSIVE DIFFERENTIALS. gI 


aA 

OR baited (2) 
CB 

ioe (3) 
HAC 

reer (4) 
rete 

Boe AL (5) 


If now we make y=a, then /(x+y) becomes F(x), which 
we will represent by z. Under this supposition equation (1) 
will become 

wz (now become z) = 4 
Substituting this value of 4 in equation (2) we have 
a 
Substituting this value of B in equation (3) we have 


whence 


similarly we have 
eee: arg 


pV le 23 ae 


and 


and so on. 
Substituting these Aas in equation (1) we have 


gz Wee 
veF (et )=245 re LEP Ms EE ees Sie 


in which the first term is what the nett becomes when 
y=o, and all the coefficients of the powers of y are derived 
from it on the same supposition. 


Q2 DIFFERENTIAL CALCULUS. 


This is the formula of Taylor. : 
A function of «—y is developed by the same formula by 
changing y into —y, thus: 
@3 as ae 
WS ENE TY) PX a Dae a etc 


EXAMPLES. 


Ex. 1. Let it be required to develop (x+y)”. 
Representing this function by z we have 
u=(x-+y") and z=x" 
then by differentiation 
26 3 
& =x, TS =aln— i) come nln 1) (2—2)xr-8 
Substituting these values in the formula we have 


=! oer ey n—1, 1") n-9,9 1 m(n—1) (n—2) » 3.9 
w(x) 8x" Ae ye ay Eg By 4 
etc., the same as by the binomial theorem. 

Ex. 2. Develop the function / x+y. 


sae 
o> 


1 1 3 5 
> > 5 1 —2 jh AP Bote 
aie: CLT ty eee oo ee 
His. (ob) SRO ae ee ey etc, 
Ex. 3. Develop ¥/ x+y 


a5 
x %y3— etc, 


+.,.-% tee re ee 


Develop the function (x—y)”. Aus. 


4 
.5. Develop the function (x—y)®. Aas. 
6 


: 1 
Develop the function 55. Azs. 


Ex. 7. Develop the function of eee Ans 

(27) Although a function of the sum or difference of two 
variables can generally be developed -by this formula, yet 
there are cases in which the coefficients (which are functions 
of one of the variables) may, by giving certain values #0 the 
variable they contain, become infinite. In such cases the 


formula cannot be applied; for in general such values for 


SUCCESSIVE DIFFERENTIALS. 93 


that variable, would not reduce the function itself to infin- 
ity, although it would have that effect on its development. 
Thus in the function 

u=a+ (x+y) 
we have 
az I igs: I 


i 3 
4(—x)* 


1 
z=at+(b—x)?, Sot Seas ere 7 = 
ax 2 (b—x)? ax 


and so on. 
If now we make x=4J, all the coefficients will become infi- 
nite, and we should have 


ij 
u=at+(b—x+y)? =a+ 0 
by the formula instead of having as we ought 


u=a ates : 
which cannot be, for the value of y is not dependent on 
that of x, and hence wz is not necessarily infinite when 
x=; but for all other values of x the formula will give the 
true development of the function. 

And herein is the difference between the formulas of Tay- 
lor and Maclaurin; when that of the former fails it is for 
only one value of the variable ; while that of the latter when 
it fails at all, fails for every value of it. 


Note.— In fact, the theorems of both Taylor and Maclaurin are founded on the 
principle illustrated in Art. 21. The real object of both is to find from the rate of 
change of a function what will be its new state arising from a given change in the value 
of the variable. 

The general method of doing this is to find the successive differentials of the func- 
tion in its first state, and then to multiply the successive differential coefficients by the 
successive powers (properly divided) of the actual change in the variable, this will give 
the actual successive partial changes in the function which together make up the 
entire change, and thus develop the function in the new state. 

For this purpose the variable must have two points of value; one where the furc- 
tion is to be differentiated, and the other, the new value produced by the change ; and 
to this end the variable, in algebraic functions, is made to consist of two parts, either 
by making it a binomial or something that may be reduced to that form. 

In Maclaurin’s theorem the variable consists of a constant and a variable combined 
together, so that their united value is a variable one, and the constant part is simply 
one point in that variable value. This is the point at which the differentiation of the 


94 DIFFERENTIAL CALCULUS, 


function is made ; but as a constant cannot be differentiated, the variable is attached 
to it long enough for that purpose and then made zero, In Taylor’s theorem the varia- 
ble is the sum or difference of two others, and the poimt of differentiation is when the 
variable has reachcd the value of one of its variable parts. This being a variable, the 
function can be differentiated directly, and the other variable may be made zero defore 
the operation. Hence the theorems of Maclaurin’s and Taylor are alike in this: both 
have a compound variable having two points of value, both are differentiated at the 
same point, and the successive differential coefficients, which are precisely alike in form 
and value, are multiplied by the successive powers of the change in value. The only 
difference is that in one the differentiation at the required point is made 7xd/rectly, and 
the variable change made zero afterwards ; while in the other the differential is made 
directly, the variable change being made zero beforehand. Hence a function of a 
binomial variable may be expamded by either method. By Taylor’s, considering both 
terms variable and reducing one to zero de/ore differentiation ; or, by Maclaurin’s, by 
considering one term as constant and reducing the other to zero after differentiation. 
Thus in the case of the function iz +y) Ls the differential coefficient will be pre- 
cisely the same if we reduce y to zero and differentiate 2 by Taylor’s method, or con- 
sider + as constant and reduce y to zero after differentiation, by Maclaurin’s method. 
In order thata binomial may represent a single variable, both terms must be subject 
to the same conditions, so that each term may be considered asa part of the same 
compound variable ; and the failing cases in Maclaurin’s and Taylor’s methods are 
simply those in which the binomial variable becomes a monomial, by giving the variable 


1 
; 3 Pet se : ; : 
a certain value. Thus the case cited in Art. 24,¢-+ QX*, isnot a true binomial vari- 


able, since the terms are not subjected to the same conditions. If we make it 


a 
(ctax)? we have a true binomial variable, and the differential coefficient 


4c 0) 2 will ot reduce to infinity when +=o. Similarly the case cited in Art. 
ie 
27, namely, 7—q +(s—x +y) * is one in which when x=4, the variable in the 


function reduces to y, and the function itself to a+y*, which does not contain a 


binomial variable of the required form. 
The same principle will apply to transcendental functions; which, in order to be 
developed, must have two points of value in the compound variable — one for the dif- 


ferential and the other for the development. Thus @”% may be expanded by Mac- 


laurin’s theorem, since it has two points of value, one at %, the point of differentiation 
where +=a, and the other the full value produced by x. 


Ce LOIN eb VW" 


MAXIMA AND MINIMA. 


(28) We have seen that when a variable changes its value 
at a uniform rate, the value of its function will in general 
vary at a rate that is not uniform. It may increase at a 
diminishing rate, until at a certain point it ceases to increase 
and begins to diminish, in which case the turning point is 
the one of greatest value, and is called a maximum. Or it 
may decrease to a certain point and, having attained its min- 
imum, begin to increase. The problem is to find whether 
there zs a Maximum or minimum value for a function, while 
its variable is uniformly increasing, and if there is, to find 
the corresponding value of the variable and its function. 

(29) While a positive function is increasing as the varia- 
ble increases, its rate of change or differential will be posi- 
tive; and negattve when it decreases (Art. 3). Hence when 
a function is passing through a maximum or minimum value, 
the sign of the differential coefficient must change from 
minus to plus or from plus to minus — the former in case of 
a minimum, and the latter in case of a maximum. 

But such change can only take place while the differential 
coefficient is passing through zero or infinity. Our first 
inquiry then is whether there is any finite value of the vari- 
able that will reduce the first differential coefficient to either 

95 


96 DIFFERENTIAL CALCULUS. 


-of these values. For this purpose we solve the equation 
formed by placing the first differential coefficient equal to 
zero, and thus find the corresponding value of the variable. 
Here we have one of three results. 

First. There may be no real value for the variable. In 
this case there is neither maximum nor minimum. 

Second. There may be a real finite value for the variable 
that will reduce the differential coefficient to zero. In this 
case there will probably be a maximum or minimum. 

Third. There may be no finite value of the variable that 
will reduce the differential coefficient to zero, but at the same 
time there may be one that will reduce it to zzfinzty. In this 
case we form the equation by placing the differential coeffi- 
cient equal to infinity, and the root that satisfies the equa- 
tion will indicate a frodable maximum or minimum. 

In order to determine in the two latter cases whether there 
7s a Maximum or minimum value of the function, and if so 
which of the two it is, we may substitute in the function, in 
place of the variable a quantity a little less, and one a little 
greater than that derived from the equation. If the result 
in both cases 1s less than when the true value is substituted 
there is a maximum; if greater, there 1s a minimum value 
of the function for the true value of the variable. 

We may also determine the same thing by substituting 
these approximate values in the differential coefficient, which 
the true value reduces to zero. If they cause the result to 
change the sign from plus to minus by substituting first the 
less, and then the greater quantity, there is a maximum, for 
the function is passing from an increasing to a decreasing 
state. If the change is from minus to plus, there is a min- 
imum, for the function is passing from a decreasing to an 
increasing state at that point. 


MAXIMA AND MINIMA. 97 


EXAMPLE. 


Find the value of « which will render z a maximum or 

minimum in the equation, 
u=x>—ox* 244 —7 

Differentiating and placing the differential coefficient 

equal to zero we have 
= 30? —182+24=3(x?—6x+8)=o0 
from which we find 
e=4 and #=2 

If we substitute in the function and in the different coeffi- 

clent I, 2, 3, 4, 5, etc., successively, we shall have for 


au 

Sie er th tee 5 
au __ 

Ma eee 3 ee 7 -O 
fe: = du __ 

Mere autae, | Ciet al) a. og F 
avis *% du __ 
gots Aad OW a 
X—=5 Sas) a OLA is st ag 
a = du __ 

R= Ole PUR 26. «oi 24 


Indicating that for x=2 the value of the function is a 
maximum, the differential coefficient passing from plus to 
minus; and for x=4 the value of the function is a mini- 
mum — the differential coefficient passing from minus to plus. 

(30) It must be understood that by maximum and mini- 
mum is not meant the absolutely greatest or least value of 
the function, but the ‘urning point, from an increase toa 
decrease, or wice versa. Hence there may be as many max- 
ima or minima of the function as there are values of the 
variable that will reduce the first differential coefficient to 
zero or infinity. 

It is also to be understood that in the discussion of ¢hzs 
subject, when a function is stated to be an zucreasing one, it 

6 


a 


98 DIFFERENTIAL CALCULUS. 


is meant that it is either positive and becoming greater, or 
negative and becoming less. If it is said to be decreasing, it 
is either positive and becoming less or negative and becom- 
ing greater. Thus if we take the function _ ; 

u=x* — 25 
and make «x, successively, equal to 

TRO a hae 
the successive values of the function will be 
24,2 1s 105-010; otek See praet 

and it is said to be increasing throughout the whole change, 


although at first its numerical negative values decrease. 
This is also indicated by the sign of the differential coeffi- 


cient which is positive as long as x is positive. The terms. 


“increasing” and “decreasing” then, in this case, refer 
merely to the a@rection in which the function is changing, no 
matter on what side of zero its value may be. 

(31) There is another method of ascertaining whether the 
first differential coefficient changes its sign on passing 
through zero or infinity, for this is the unfailing test of a 
maximum or minimum. Having found that value of the 
variable which reduces the first differential coefficient to 
zero, substitute that value in the second differential coeffi- 
cient, if it contain the variable, then 

first, Tf it reduces the second differential coefficient to 
a negative quantity, it indicates that when the first is at zero 
it must be a decreasing function, which can only be at that 
point by its passing from a positive toa negative state, and 
hence the function itself must be passing from a state of 
increase to a state of decrease, and hence is at a maximum. 

Second. If it reduces the second differential coefficient to 
a postive quantity, it indicates that the first when at zero is 
an increasing function, and must, therefore, be passing from 
a negative toa positive state, hence the function is passing from 


a 


MAXIMA AND MINIMA. ai 99 


a decreasing to an increasing state, and is, thx fore, at a 
minimum. | 

Third. If it reduces the second differential coefficient 
zero, We may resort to the third; and if the same value of 
the variable reduces that to a real finite quantity, either pos- 
itive or negative, it shows that the second, at zero, is chang- 
ing its sign, and, therefore, the first is changing from an 
increasing to a diminishing function, or we versa, and, 
therefore, does not at the zero point change tts sign. Hence 
there is neither maximum nor minimum in the value of the 
function. 

Fourth. Tf it reduces the éAzrd differential coefficient to 
zero we may resort to the fourth. If it reduces this to a real 
finite value, it indicates that at zero the third changes its 
sign, for it can only increase on both sides of zero by pass- 
ing from negative to positive, or diminish on both sides by 
passing from positive to negative. This will show that the 
second coefficient does zo¢ change its sign, for if it increases 
on one side of zero’ and decreases on the other, or wice versa, 
it can only approach the zero point until it éowches 77, and then 
must recede without changing its sign. This proves that the 
first coefficient does change its sign, for since the second does 
not change the first must be passing ‘Arough from one side 
to the other. There will, therefore, be a maximum or mini- 
mum —the first if the fourth differential coefficient has a 
negative value, and the second if it is positive. 

We may continue thus and show thatif the first differen- 
tial coefficient that is reduced to areal value, by substituting 
that value of the variable that reduces the first to zero, is of 
an even order and postive, there will be a minimum ; if it is 
negative, there will be a maximum; and if it is of an odd 
order there will be nether maximum nor minimum. 

Fifth. If any value of the variable reduces the first dif- 
ferential coefficient to infinity, it will probably reduce all the 


Too DIFFERENTIAL CALCULUS. 


succeeding ones to infinity, also. It will, therefore, be best 
in such a case to substitute values for the variable a little 
less and then a little greater than the one found. If the 
value of the first differential coefficient changes from plus to 
minus there is a maximum, and the second will be plus on 
both sides of infinity ; for the first must be an zzcreasing pos- 
itive function in order to become positively infinite, and if 
negative on the other side must be a decreasing function, 
for it cannot be an increasing negative function on leaving 
infinity. Hence (Art. 3) the second must be positive in both 
cases. 

Sixth. If the first differential coefficient in the last case 
changes from minus to plus there will be a minimum, and 
the second coefficient will be minus on both sides of infinity. 
Thus we see that when any value of the variable reduces 
the first differential coefficient to zero, and is: substituted 
in the second, a mnus result indicates a maximum in the 
function, and a plus result a menimum. When any value 
reduces the first coefficient to zzfinity, a plus sign for the 
second indicates a maximum, and a minus sign a minimum. 


EXAMPLES. 


fix. t. In order to illustrate the first case in this article 
we take the function 


u=16x—x? (1) 
from which we obtain by differentiation 

Tut 

Tp 16 28 (2) 

a*u 

i (3) 


We find that x«=8 will reduce the first differential coefficient 
to zero, while the sign of the second is minus. Hence at 
x«=8 the first must be a decreasing function, and, therefore, 


MAXIMA AND MINIMA, IOI 


passing through zero from plus to minus, the function will, 
therefore be an increasing one to that point and then a 
diminishing one; hence a maximum. 

If we substitute in the function and the first differential 
coefficient for « values a little less and a little greater than 
8, we have for 


du 
x=7 uw=63 ig 2 
C= 5 uz—64 eo 
—_ saat du 
x—9 eee u=63 a hac ap 


If we represent the values of the function by the ordi- 
nates of the curve ABC (Fig. 3), the curve itself will cor- 
respond to the range or locus of values of the function, while 
the variable increases uniformly in passing from 7 to 9. 
From A to B the function increases, B 
but at a decreasing rate, and hence “ C 
the first differential coefficient is 
positive but decreasing until it 
reaches zeroat B. The function then 
decreases at an increasing rate, and 
hence the first differential coefficient 7 P 9 
must be negative and increasing. Fig. 3. 
But when this coefficient (or any variable) is positive and 
decreasing, or negative and increasing, z¢s rate of change, 7.¢., 
the second differential of the function, must (Art. 3) be neg- 
ative throughout, which corresponds with the result found in 
equation (3). 

fx. 2. To illustrate the second case we take 


u=x*—16x+70 (1) 
from which 
Lu 
Pei AS (2) 
£ Gite 


Watts (3) 


102 DIFFERENTIAL CALCULUS. 


° . du - . . 
We infer from equation (3) that > is an increasing func- 
tion for all values of x, and hence it is so when x=8, which 
du . : du + : 
reduces 7, to zero. From which we infer that > 1s passing 


from a negative to a positive state, and the function itself 
from a decrease to an increase. Hence a minimum. 

If we substitute 7, 8 and g successively for x in equations 
(1) and (2), we have for 


ask os Gitte 
a, ee cee 
aes Solas du 
Vo sao ee, 
du 


which corresponds with our deductions. 
If we let the ordinates of the curve A BC (Fig. 4) rep- 
resent the values of w, we see that from A 


A to B the value of wz diminishes, as is B : 
shown by the sign of “ oo and at a dimin- 
ishing rate as is shown by the positive . 

pee) 7 on nis. 
sign of [a * (Art. 3). From. Bto-C zw Fig. 4. 


increases, as is shown by the positive sign of = =, and at an 


Date . : 
increasing rate, as is shown by the sign of ee, which is 


still plus. Hence the shape of the curve. 
Ex. 3. To illustrate the third case we make 


uw=9+2(x—3)8 (1) 
whence 

Au 

w= 6(x—3)? (2) 

au 


=12(x—3) (3) 


Here we find «=3 reduces - to zero, and hence if there 


axe 


is a Maximum or minimum it will be for that value of «x. 


MAXIMA AND MINIMA. 103 


9° 
ww 


au 
But it also reduces 7g Tee to zero also, hence we resort to the 
aru f ; ; 
value of xe? which we find to be 12. We infer from this 


a*u ee : 2 ee 
that when Fee =O it 1s passing from negative to positive, 


hence § = ~ is passing from a decreasing to an increasing func- 


tion at ae zero point, and, therefore, does zot pass through 
it. It is, therefore, all the time positive, and the function is 
at all times an increasing one, so that there is neither max- 
imum nor minimum. We may, also, learn the same thing 


du 
from inspection, for since the value of > 1s a square it must 


always be positive. 
If we substitute in the given function and in the values 


du au 


of ow and re the numbers 2, 3 and 4 successively for x, 
we have for 
du a? u 
SEARS Capel hay ed 0 OC aa 
Lu ae 
x= “= > = FiO 
3 Pana aa? 
du Ce 
Cee AS Bs Tha re =6 Seba te 


If we let the ordinates of the curve A B C (Fig. 5) repre- 
sent the values of wz, we see that from 2 to 3 the function 
eeeesgaes as is shown by the positive C> 


value of ~, 


asis indicated by the negative sign of 


au 
ere (Art. 3), between those points 


but at a decreasing rate, 


or while x is less than 3. From 3 to 
4 the function is still increasing, as is 


104 DIFFERENTIAL CALCULUS. 


shown by the positive sign of 7 and at an increasing rate, 


eA e a* u 
as is shown by the positive sign of aya when the value of 


du 


xis greater than 3. Hence at B where the value of 7 is 


zero, the function having increased at a decreasing rate up 
to that point, ceases for an inappreciable moment, and 
begins again to increase at an increasing rate. 

ix. 4. To illustrate the fourth case take 


u=53+(e—7)* (x) 
whence 

F = 4(4—7)3 (2) 

Tt x47)? (3) 

Sraee— (4) 

Chang (5) 


Here we see that the fourth differential coefficient is the 
first that has a real finite value for x=7, which reduces all 
the preceding ones to zero. Hence, according to our rule, 
there should be a minimum value for the function at that 
point. In fact, the sign of this coefficient shows that the 
third changes at zero from minus to plus; and this shows 
that the second does zof change its sign at zero, but after 
being a decreasing function to that point, becomes an in- 
creasing one, and is, therefore, positive both before and after. 
And this again shows that the first des change its sign at 
zero, since it is an increasing function on both sides of zero, 
which can only be by passing from a negative to a positive 
value. Hence the function will decrease until x=7, after 
which it will increase, showing a minimum at that point. 

If we substitute in the given function and in the differen- 


MAXIMA AND MINIMA. 105 


tial coefficients the numbers 5, 6, 7, 8, 9 successively for x 
we shall have for 


Lu ape Wate 
ee aol Lei ig 2a era =48 ear 
x=6 w=6 “=—4 “ =12 “ =—24 
x=7 u=5 “=o “ =o “* =o 
C6 . =. Oe — “ =12. “ =24 
x=9g uw=11 “ =32 “ =48 %“ =48 


which illustrates the conclusions we have drawn. 
The general proposition enunciated in the fourth case may 
be demonstrated analytically as follows. Let us suppose 
u= F(x) 
and let the variable x be first increased and then diminished 
by another variable %; and let these new states of the 
function be represented by z’ and w’, then we have 
ul’ =F (x+h) 
u" =F x—h) 
Developing these by Taylor’s theorem we have, after sub- 
tracting the original function z, 
P SrOaIey Ce een i 
Le SH aay (in mee ary, Tg OE Bi ie etc: 


P é ae au Ups Ou hs 
u mete rl reat f 


+ etc. 


Bete oar 

Since the powers of / increase in each successive term of 
this development, we may reduce the value of 4 to such an 
extent that the value of any one term shall be greater than 
that of all the succeeding terms added together. Such in 
fact will be the case if % is less than one-half in the series 
AES /L? GUC. 

Let us suppose # to be so reduced, then if w is a maxi- 
mum, it must be greater than # or wz”, and the second mem- 
bers of both these equations must be negative; if it is a 
minimum it must be less than z’ or zw”, and in this case the 
second members of both equations must be positive. 


106 DIFFERENTIAL CALCULUS. 


Hence, in case of a maximum or minimum, the second mem- 
bers of both equations must have the same sign, and the 


first term of each (which controls the value of all the rest) 


: : 2 OIE 
having contrary signs must reduce to zero; that is, — must 


> ax 
be zero, since “is not. This then is a necessary condition 
to a maximum or minimum. If there is a real value for the 
a" u 
ax* 
since #? is positive), will now control that of the whole sec- 
ond member, and will determine whether zw is a maximum or 


second term in each equation, its sign (or that of 


Lu 
minimum. If the second term (or x2 ) become zero, there 


can be no maximum nor minimum unless the third term (or 
Dead 
axes 
it has contrary signs in the two equations. We see then 
that the conditions of a maximum or mimimum are: jirsé, 
that the first differential coefficient should become zero; 
and, second, that the first succeeding differential coefficient 
that has a real value should be one of an even order, since the 
even terms have the same sign in both equations. If that 
is negative, the whole of the second member of the equa- 
tion is negative, and there is a maximum; if it is positive, 
there is a minimum. Which agrees with the rule already 
found. 
“x.5. To illustrate the fifth case we take 


) which is now the controlling term, is also zero, since 


2 
u=10—(x—3)% (1) 
whence 
au —2 
we 1 (2) 
*.~ 3(e=3)8 
a2 u 2 
ax? (3) 


MAXIMA AND MINIMA. 107 


Here we find that «=3 will reduce o to infinity. Refer- 


: au : 
ring to the value ot — 3 we find that it reduces that also to 


infinity, but we see by inspection that any other value for 


9 


ecere 
x, whether greater or less than 3, will make that of ee 


positive. We see also that = ~ will be positive when ~ is less 


than 3, and negative when it is greater. From all this we 
infer that the function is an increasing one before x=3, and 
a decreasing one afterwards. Hence there is a maximum 
at that point. 

If we substitute for x, in equations (r), (2) and (3), the 
numbers 2. 3. 4 successively, we have for 


_ Z, tif w bite 
toe Seopa et yet 
Ls es Qu ae ee 
heat O Beet OS org 
= #9 Cie Mn Coon 
See ye es Nee 


If we let the ordinates of the curve A B C represent 
the successive values of w (Fig. 6) we 
see that from 2 to 3 the function in- 
creases, as is shown by the positive 


value of “ *, and at an increasing rate 


as is shown by the positive value of 


(hte 
fomai: m 3 to 4 the negative sign 
Tne From 3 to4 the negative sig Baer 


of = indicates a decrease of the function, while ale positive 


mien 
sign of 2 = (which has not changed) shows that this decrease 


lsat a Fans rate. Hence the form of the curve. 
Lx. 6. To illustrate the sixth case we take 


108 DIFFERENTIAL CALCULUS. 


2 
u=(3x%—9)* (1) 
whence 
Lub 2 (2) 
i ae 4 4 
(3~—9)* 
es 2 (3) 
Hite OO 4 3 
(3-9) 
In this case, as before, we find that x=3 reduces ~ and 
au : : F , Cu 
ee ' infinity. We see also by inspection that a changes 


from minus to plus as x passes from less than 3 to greater, 


9 
4 


axe” 
infer that z is a decreasing function until «=3, and an in- 
creasing one afterwards. Hence a minimum at that point. 

If we substitute for x in equations (1), (2), (3), the num- 
bers 2.3.4. successively we shall have for 


while is negative on both sides of infinity. Hence we 


ge UU Demme ale 2 
x=2 “u=} > =— 3 SS eee 
/ 9 ax R/ 3 ax” A/ I 
bet of aa Oe 
x=3 u=o ie Te wie 
- Ses ae Zeus 2 
Cig tee a V Weary x/ 53 28 SOR cars 
If -we let the ordinates of the curve A B C (Fig. 7) rep 
resent the successive values of z, A | 
we see that from 2 to 3 the function ; 
decreases, as is shown by the nega- 
tive value of =, and at an increas- 
ing rate, as is shown by the negative | 
a* 4 2 3 4 
l anes 
value of 7x2 > from 3 to 4 the pos Fig. 7. 


we du - e e = ° : 
itive value of 7 indicates an increasing function, while the 


MAXIMA AND MINIMA, Iog 


. . a . . . 
negative value of FEES shows that increase to be at a dimin- 
ishing rate. Hence the form of the curve. 

(32) We have seen (Art. 30) that there may be as many 


maxima and minima of a function as there are roots for the 


equation formed by making the value of =e. To illustrate 


a case of this kind we take 


ERE gs 
u=x* —20x%3—132x%?—320x 4286 (1) 
whence 
wt 
Tn 4k? — 60x? + 2644— 320 (2) 
au 
Tuk 12K? — 1200+ 264 (3) 


Placing the second member of equation (2) equal to zero, 
we find for x three values as follows : 
a= 2 
By a 
—o 
Substituting these values in equation (3) we have for 
Gas 
X=2 “3-7? 
Cu 
eae ax® oe 
lect 
axe 1? 


from which we infer that for x=2 and x=8 there is a mini- 
mum, and for x=5 there is a maximum. This will be seen 
by substituting for x in equations (1), (2), (3), successive 
values, as follows 


I10 DIFFERENTIAL CALCULUS. 

Lu aul 
= 0 W286 Wen oto. yt ee 
c= 1 @eleg -lio= tiie eee 
we Wa eR Ole dae OU Phere 
x= 30 uw 55 NS yo ae 
4S 4. w= 64 1S Sl ee Pee 
x= 5 w=111 “ = o 6 =—36 
x= 6 u= 94 “ =— 32 “ =—24 
ee Reg SO eee Ot a) eS 
2 (8) y= 300 eee Rest 
Kg v9. PS are rb 
KAO 2 OO eee S20 eo 


If we let the ordinates of 
resent the successive values 
wz corresponding to numbers 
substituted for x at the foot of 
each, we see that the function 
decreases at a diminishing rate 
until «=2, when it ceases to 
decrease and begins to increase 
at an increasing raté, as 1s 


the curve ABC (Fig. 8) rep- 


fs 


of | 8B 
Ht 
fo 2°S Fee LOT eae 
Fig. 8. 


Wi ; 
— and the continued 


shown by the change of sign in oy 


posi- 


a u 


tive sign of Wee" But at «=4, although still increasing, as 


fy th ee, eee 
is shown by the positive sign of yu? it is at a diminishing 


a u, 
Tae 8 how negative, and thus continues until at 5, 


rate, for 


au 

aes becomes zero, the function ceases to increase and be- 
gins to diminish at an increasing rate, as is shown by the 
au 


ax? 


Cu 


negative signs of ye and at x=6, But at x=7 we 


a+ 


MAXIMA AND MINIMA. LE. 


” Sue 
76h he 
the function has ceased to increase and begins to diminish, 
until at «=8 it has become zero, when the changes at x=2 
are repeated. We notice that between x=3 and x=4 the 


9 
~ 


ax” 


have positive, showing that the rate of diminution of 


sign of changes from vlus to minus, showing that be- 


ut 
tween those two points the rate of on has changed from an 


increase to a decrease, that is, the function has changed 
from increasing at an increasing rate to increasing at a 
diminishing rate. The exact point where this change takes 
(Seoga a, Reoeee 
place is where the value of eae Gs This will give 
x=s—1/3 and x=5+Yv 3 

which last value corresponds to a point between «=6 and 
x=7, where the same change is repeated, only in a contrary 
direction. | 

From all these cases we deduce the following rule for 
ascertaining the values of the variable that will produce a 
maximum or minimum value for the function, if there be 


any. 
Place the first differential coefficient equal to zero; and 
substitute each of the roots of this equation for the varia- 
ble in the second differential coefficient. Each one that 
reduces it to areal negative quantity will produce a maxi- 
mum value for the function; while a similar positive result 
will indicate a minimum. Should any real root thus found 
reduce the second differential coefficient to zero, substitute 
it in the third, fourth, etc., successively, until a real finite 
value is found for some one of the coefficients. If the first 
thus found be of an even order and positive, there will be a 
minimum; if negative there will bea maximum. If the first 
that is reduced to a real finite quantity is of an odd order, 


I1I2 DIFFERENTIAL CALCULUS. 


whether positive or negative, there will be neither maximum 
nor minimum. 

The first differential coefficient may also be placed equal 
to infinity, and if there be any real finite roots, they may be 
treated in the same manner as those obtained by placing it 
equal to zero. In this case, however, a positive sign of the 
second differential coefficient indicates a maximum and a 
negative sign a minimum. 

If a given function contain two variables there must be 
an equation, and one of the variables must be considered as 
dependent on the other. The problem will be to find the 
maximum or minimum value of the dependent variable; for 
which purpose it must be considered as an implicit function 
of the other, and the differential coefficients will be found 
as in other cases. 


Nore.— It may be objected that, herein, the subject of maxima and minima has 
been treated in too prolix a manner, and the reasoning has been unnecessarily repeated. 
I reply, it is of the highest importance that the student should have not only a clear 
and correct, but a falar, conception of the Zaws which govern the relations of the 
different orders of rates or differentials, because these are among the fundamental ideas 
of the calculus, and essential to a complete comprehension of the subject. Now unless 
these ideas are presented sufficiently often to render them familiar ; if the student on 
every new occasion is obliged to draw afresh upon his powers of imagination, and go 
through the mental labor of forming his conceptions anew, the study will prove not 
only more difficult, but far less attractive. He will be like a traveler in the dark, who, 
instead carrying a constantly shining lamp to guide his footsteps, must light his candle 
anew for every fresh obstacle, Hence the importance of a full and elaborate explana- 
tion, even at the expense of some, otherwise unnecessary, repetition, 


EXAMPLES. 


Find the value of « for the maximum or minimum value 
of z in the following equations : 


Ex. 8. “w=x>+18x" +1052. Ans. 
Lx, 9.. “=a—bze-+x*. Ans. 
Lx.10. uw=at+h3x—cx?. Ans. 
Lx. 11. u=3a*x?—btx+c?. Ans. 


Lex. 12. u=a* +bx*? —cx3, Ans. 


MAXIMA AND MINIMA. Il3 


APPLICATION TO PRACTICAL PROBLEMS. 


(33) In order to apply the rules for determining maxima 
and minima of functions to the solution of practical prob- 
lems, it is necessary to obtain an algebraic expression of the 
function, whose maximum or minimum is to be determined, 
in such terms that it shall contain but one variable. No 
specific rules can be given for this purpose, but care must 
be taken to express the function in terms of a variable that 
shall have a range of values deyond that which may be 
required to produce a maximum or minimum, for if it does 
not, although there may be a kind of maximum or minimum, 
it will not be one in the meaning of the term as used in the 
calculus, as there will be no ¢urning potntin the value of the 
function, nor any change of sign in the value of the first 
differential coefficient. A few examples will indicate the 
nature of the process more clearly. 

Ex. 1. Divide the quantity a into two such parts that 
their product shall be a maximum. 

Let x be one of these parts, then the other will be a—x, 
and the function will be 

x(a—x)=ax—x? 
which is to be amaximum. Representing it by z we have 


Lu 
pn ae (1) 
aD: 
FM rae (2) 
. au 
Placing the value of oe equal to zero we have 
= @ 
eee 


9 
~ 


fhe 
Hence when a guanttty ts divided into two parts thetr product ts 
a maximum when they are equal. 


which the negative sign of shows to be a maximum. 


TI4 DIFFERENTIAL CALCULUS. 


£x. 2. To find the greatest cylinder that can be inscribed 
in a given right cone. 

Let the height SC (Fig. 9) 
of the cone be represented by 
a, and the radius of the base 
AC by 4, and let x represent the 
distance SD from the vertex of 
the cone to the upper base of 
the cylinder. From the trian- 
gles SAC and SED we have 
rn eID ED Orngs mia 
ED, hence 

ae 

But the area of the upper 

base of the cylinder is 


b* x? 
tH Ds 
a 


Multiplying this by DC=a—x, the height of the cylinder, 
we have the volume or capacity which we will represent by 
V, and hence | 

af cae 
V=— 3 x" (a—2) 
Now any value of x that will render «*(a@—x) a maximum 


mb” 
will render any multiple of it also a maximum, hence 


being a constant factor may be disregarded in the operation. 
Differentiating twice and representing x*?(a—x) by w, we 
have 


Lu 
; Me 2k 3X" 
2a , 


Making the value of 2 equal to zero we have 


Qa 
pra x=F 


MAXIMA AND MINIMA. IT5 


The first cannot be amaximum since it reduces the value of 
a” u oak oe Aid 

xe tO 24; which being positive indicates a minimum. In 
fact, when x=o the cylinder is reduced to the axis of the 
cone, and vanishes with x. The other value x=4a will 


° 


7 


¢ 
solve the problem, since it reduces the value of S=> ue 19 — 24; 


a negative quantity which indicates a maximum ee for the 
function. Hence 

The maximum cylinder that can be tnscribed in a right cone ts 
one in which the height of the cylinder ts one-third of the height 
of the cone. The radius of the base will also be equal to 
two-thirds that of the base of the cone. The volume of the 
cylinder will be to that of the cone in the ratio of their 
bases, or as 4 is to 9. 

Lx. 3. Required to determine the dimensions of a cylin- 
drical vase, that will contain a given quantity of water with 
the least amount of surface in contact with it. 

Let v represent the given volume of water, x the radius 
of the base of the cylindrical vase, and y its alee Then 
we shall have 

V=Tx"*y 
from which 


7pé 


4 iam > 
mm ane 
ve 


Now the convex surface of the cylinder is equal to 27xy, 
and substituting the value of y we shall have the convex sur- 
face equal to 

2TXU 20 


Tx? x 


If to this we add the surface of the base =zx? we have the 


whole surface in contact with the water. Calling this surface 
S we have 


116 DIFFERENTIAL CALCULUS. 


2U 
——— 4 x2 
esree geo (1) 
as 2U 
——_ = ——,- +27 2 
ax ve art (2) 
20 LAD 
aes (3) 
Placing the value of = * equal to zero we have 
27x =2y 
whence 
eA Jal 
xX — 


This value answers to a minimum, since it renders the value 


9 
o 


Ss pes ; : : 
of x2 Positive. If we substitute this value of x in the 


expression we found for the value of y, namely 


we have 


hence the minimum surface will be in contact with the water 
whet thers cht of the cylinder ts equal to the radius of the base. 

Ex. 4. \ It is required to inscribe in a sphere acone which 
shall have the greatest convex surface. 

Suppose the semi-circumference AMB (Fig. 10) torevolve 
“about the axis AB, it will describe M 
the surface of a sphere, and the 
chord AM will describe the con- 
vex surface of a right cone in- 
scribed in the sphere, and AP will 
be its height, and PM the radius 
of the base. The convex surface 
of the cone, which we will call S, “9 
will be iets 


MAXIMA AND MINIMA. ERT 


S=27PM .4AM=zPM.AM (1) 


We have now to determine PM or AM, either of which 
will determine the other. 


het 
AB=z2a and AP=x 
then rai 
PM =AP. PB=x(2a—2) 
and 
PM=V a(2a—2) 
Again 


AM=\/ 20x 
Substituting these values in equation (1) we have 
=n7zPM. AM =27v/ 20x —x2 . / 2ax=7/ 4a? x? —20x8 
Differentiating this function we have 


ee eae ak AGT — 300 


== a 2 
Lx A/aarx2 —2ax3 ~/ 4a2—2ax ( ) 
= segs ds 
Placing this value of |, equal to zero we have 
— 4a 
ears: 


In order to determine whether this value of x corresponds 
to a maximum or minimum of the function, it will be neces- 
sary to find the sign of the second differential coefficient. 

Before doing this we will examine a method by which the 
Operation may be somewhat abridged. 

We have already seen that when the value of a function 
is reduced to zero by giving a particular value to the varia- 
ble, it does not follow that its differential will also be reduced 
to zero by the same value of the variable (Art. 31), for the 
function in passing through zero may, and probably will, be 
passing from negative to positive, or we versa, and, there- 
fore, may have a differential or rate of change at that point, 
the same as at any other. Thus the latitude of a vessel on 
the equator is zero, but it may be changing as rapidly there 
as anywhere else. 


118 DIFFERENTIAL CALCULUS. 


Now if we wish to obtain the second differential coefficient 
for a particular value of the variable, we may take advantage 
of this circumstance. Suppose we find the first differential 
coefficient to be the product of two or more factors; either 
of these being reduced to zero will reduce the coefficient to 
zero. In this case we may obtain the value of the second 
differential coefficient for the corresponding value of the 
variable, without differentiating the entire coefficient. For 
suppose we have 


- Ss 
in which y. y’ and y” are functions of x ; this product will 
be reduced to zero by any value of x that will reduce either 
factor to zero. Suppose that for x=a we have y=o ; if we 
differentiate the function we have 


Uae) a? u_ayy'y")_ sy" dy yy a! yy ae 
th aa Dee TR ae. op eee ine 
and since x=o reduces y to zero, the two last terms of this 
expression become equal to zero, and we have 
etn Vt ae 
ax® ax 


ae 
hence to obtain the value Ot as 3 Jor that value of x that 


reduces one factor of the first Se res coefficient to zero, 
we have only to multiply z¢s differential coefficient by those 
factors which do zof become zero, and then substitute the 
value of x. If for example we have 


a x(x? —a?)=x(x+a)(x—a) 
we may reduce it to zero by making 


B= 0,0 CeO ee 
a1 
If we wish the value Wereae De® SF the first value of x, we have 


au 


———s — 72 2— 2 
9 —X*" —a” ——a 
aX” »—0 


MAXIMA AND MINIMA. 119 


If for the second value we have 


au ft : 

Ti pag to0) =20 
If for the third value we have 

au 

—— a —q)=2q? 

SEE Ns x(x—a)=2a 


Resuming now equation (2) 
aS. 40" — 3ax a ' 
ax x/4aX®—2ax WW 4a2 — 20x 4a—3) 


4a 
—. and hence 


which becomes zero by making 3x=4a or x=3, 


Jor that value of x we shall have 
CBee a ae Nal ere ae 


ax® — a/4a® — 20x ° ax ir a/4a® — 2ax 
. . ° 4a 
which being negative shows that x=, corresponds to a max- 


imum of the function. 

Hence, zf a right cone be inscribed in a given sphere tt will 
have the greatest possible convex surface when the axis of the 
cone ts equal to two-thirds of the diameter of the sphere. 

Ex.5. A point o (Fig. 11) being given within the right 
angle B AC, through which 
a line is tobe drawn meeting p 
the axes AB and AC, it is 0 
required to find the distance 
Az such that the length of 
the line between the points a a al! 
of its intersection with the Fig. 11. 
axes shall be a minimum. 

Let Am=a, Om=6 and mma=x. Then the right angled 
triangles Om and pAzn give the proportion mz: Om:: An: Ap 
or 


C 


pO LEAS! Sa ae OS 
whence 


I20 DIFFERENTIAL CALCULUS, 


But ; 
hp en — oa 
whence 
_ 2 b(atx)?* 
gn EE yee 
whence 


a+x 
pa= Tere +42 


which is the function of which we are to find the minimum 
value. 

Representing this function by wz, and considering it as the 
product of two factors, we have 


Xx fe Ronit 
= a ————— ie 
ax K APSR ee TV. DS Hat ore 
Reducing to a common denominator we have 
dé (ae) x* ab? fa"), ee ee 


ax x 4/ $2 4x2 £8 N/R +2 
Making this equal to zero we have 
x= h/ab2 


To find if this is a minimum we differentiate again, but 
as the numerator of the differential coefficient is equal to 
zero for this value of x (Ex. 4), we multiply its differential 
by 

I 
X4/ 52 +H? 
which gives 
eh FOO): Rae 7 2 
dx? — x 4/2 4x2 4/52 4x2 


a result that is essentially positive whatever may be the 
value of x. Hence x=X/as? corresponds to a minimum 
length of the line gv, If a and 4 are equal, we have 


MAXIMA AND MINIMA. r21 


x=6 or mnu=Om 
whence 
Ap=An 
£x.6. To find the maximum rectangle that can be in- 
scribed in a given parabola. 
Let A C B (Fig. 12) be 
the parabola of which 
1). is the, axis’ Let 
DE be the height of the ae 
inscribed rectangle, and 
let CD=a and CE=x, 
then FE? = 2fx and 
FG=2,/2px. ‘ 
The area of the rec- 
tangle is 


Fig. 12. 
FG xX ED or 24/24x(a—<x) 
which is to be a maximum. 
Dropping the constant factor 24/ 2 (Ex. 2), representing 
the function by w and differentiating, we have 


deere. au =) 1 
u=ax*—x* and >-=tax *—3x* 
ax ¥ 
which being made equal to zero gives 
x= a 
te 
hence che altitude of the rectangle 1s equal to two-thirds that of 


the parabola. 
To show that this is a maximum we differentiate again, 


and find 


oo a fa 
i ge ee 


which is negative for every positive value of x, and therefore 


ao 


for x=. 
Ex.7. What is the length of the axis of the maximum 
parabola that can be cut4rom a given right cone ? 


122 DIFFERENTIAL CALCULUS. 


Let ABC (Fig. 13) be the given cone, C 
and FDH the parabola cut from it. \ 
Let DE be the axis of the parabola, y 
and AB the diameter of the base of 
_the cone. Represent AB by a, AC by 
6é,and BE by «. Then AE=a—z, / 
FE=V/ax—x?. Also 
ADAG? EBink Dora .v. we pap rf B 
hence eae 
ED ne 13. 
But the area of the parabola is eee to 
$FH x DE or § — . 20/ax—x? 


which is, pea to be amaximum. Dropping the con- 
stant factor 5 = (Ex. 2), representing the function by wz, and 


cneeeecn ne we have 


U=A/ ax3 —x4 (1) 
Gt AAI ee At I 
BYP Sere ples 
ax 24/ Axe — x4 apy rs Tare . (3ax? 4% ) (2) 


Placing the second member of equation (2) equal to zero 


=". and differentiating (2) for this last 
value of x we have (Ex. 4) 

hes I 

ax* ~~ 2r/ax8—x4 * 
Substituting this value of x in the second member of this 
equation it is reduced to 
fee he 
ee 2/5 


we have x=o andx 


(6ax—12x") (3) 


flence the axts of the maximum parabola ts three-fourths of 
slant height of the cone. 

“ix. 8. It is required to determine the proportion of a 
cylinder, that shall have a given capacity, and whose entire 
surface shall be a minimum. 


MAXIMA AND MINIMA. 123 


Let a? be the capacity of the cylinder, x =4AB (Fig. 14) 
the radius of the base, and y=AC 
be the height; then the two bases c D 
taken together will be equal to 27x?, 
and the convex surface to 27xy, so that 


am(x*?+xy) is to bea minimum. Now 
3 


a : 
zx*y=a>, whence Sense: Substitu- 


ting this value of y, representing the 4, - 
function by wz, and differentiating we 


have Fig. 14. 

2a 
a x pie Bin 

uso? +— (1) 
au ane 

Te TARE — a (2) 

a? u Aan 
ax —4 x4 (3) 


Placing the value of o equal to zero we have 
atxs =ae=rx*y 
whence 
y=2x 
or the height of the cylinder 1s equal to the diameter of the base 
and 
om 
V on 
This value of x corresponds to a minimum value of the 
function, as is shown by substituting it in equation (3), 
which gives 


a positive quantity. 

Ex.9. ‘To divide a right line into two parts such that 
one part multiplied by the cube of the other shall be a max- 
imum. 


Ans. The part cubed is three-fourths of the given line. 


124 DIFFERENTIAL CALCULUS. 


x. 10. To find the greatest right angled triangle that 
can be constructed on a given line as a hypothenuse. 
Ans. The triangle must be isosceles. 


Ex. 11. It is required to circumscribe about a given par- 
abola, a minimum, isosceles triangle. What is the length of 
its axis? 

Ans. ¥Four-thirds the axis of the parabola. 


Ex. 12. What is the altitude of the maximum cylinder 
that can be inscribed in a paraboloid. 
Ans, Half that of the paraboloid. 


£x. 13. The whole surface of a cylinder being given, 
how do the base and altitude compare with each other when 
the volume 1s a maximum ? Ans. 


fx. 14. Required the minimum triangle formed by the 
axis, the produced ordinate of the extreme point, and the 
tangent to the curve of a parabola. Ans, 


Py Hea OuNes Ve. 


PRPLICASION OF VI HE DIFFERENTIAL CALCULUS TO 
PI tee Ore y ROL Ge URL 


SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT. 


(34) In order to form such a conception of a line as will 
be adapted to the methods of the differential calculus, we 
must consider it to be “le path of a flowing point. 

The Zaw which governs the mcvement of the point deter- 
mines the nature of the line, as to forrn and position; and 
this law is expressed in the Cartesian system by the eguaton 
which shows the relation between the co-ordinates of the 
generating point in every position it may occupy throughout 
its movement. 

The drection in which the point is moving is determined 
by the relative rates at which the coordinates are changing 
their values at the same moment. If the rate of change 
should be constantly the same in each of the coordinates, 
whether negative or positive, the generating point would 
move constantly in the same direction, describing a straight 
line; and this direction would be determined by the razo of 
these rates, which in this case would be measured by the 
simultaneous increments or decrements of the coordinates 


themselves. 
125 


126 DIFFERENTIAL CALCULUS. 


If, for instance, the coordinates AB and BC (Fig. 15) have 
each a constant rate of increase, 
the ratio of the increments CO and 
OC’ will be constant, and the gen- 
erating point C will move in a 
straight line, whose direction will 
be determined by the relative rates 
with which those increments are 
produced ; or since the rates, being 
uniform, may be represented by 
the simultaneous increments, the 


direction will be determined by the ratio ae 


But if, while the rate of change in one of the coordinates 
is constant, that of the other should constantly vary, the 
ratio of their simultaneous increments would be constantly 
changing and the point would describe a curve, whose char- 
acter would be determined by the Zew which should govern 
these varying rates of change; and /Azs law would be expressed 
in the equation of the curve. But if the varying rate of change 
in the coordinate should, at any point in the curve, cease to 
vary, and should continue afterwards constantly the same 
as at that point, the generating point would cease to describe 
a curve, and would move in a straight line in the direction 
to which it was chen tending; and this direction would be 
determined by the ratio between the rates of change in the 
coordinates as they existed at the instant they both became 
untfor me. 

Low the tangent to any curve ts the line which would be des- 
cribed by the generating point if tt were to move in the direction 
to which tt ts tending on tts arrival at the point of tangency ; 
just as a stone, when it leaves a sling, describes a line tan- 
gent to the curve in which it was moving at the instant. 
Hence the ratio of the rates of change in the coordinates of 


THEORY OF CURVES. 127 


any point of a curve determine the direction of the line 
tangent to the curve at that point. 

Suppose, for example, that AB (Fig. 16) has a uniform rate 
of increase, while BC has a rate 
of increase that is constantly 
diminishing, the point C would 
describe a curve. 

Now let us suppose that the 
generating point on arriving at the 
point C of the curve should con- 


tinue to move in the direction 5 ie ea Bt 
towards which it was then tending, Aas 

and should with a uniform motion, at the same rate as it had 
at C, describe the right line CD, then this line would be tan- 
gent to the curve at the point C. From D draw a line DB’ 
parallel to the ordinate, and meeting the axes of abscissas at 
B’; and from C draw a line parallel to the axis of abscissas 
meeting DB’ in O. The triangle CDO will have important 
properties which will require careful investigation. 

From whatever point in the line CD the line DB’ is drawn, 
the ratio between the lines CD, CO and DO will be the 
same, and hence as CD is described at a uniform rate, equal 
to that with which:the generating point 1s moving at the 
point C, the lines CO and DO will also be described at a 
uniform rate equal to that with which AB and BC were in- 
creasing at the same instant. Hence the three lines CD, 
CO and OD are the increments that would take place in the 
arc, the abscissa and the ordinate in the same unit of time, at 
their several rates of change existing when the generating 
point of the curve is at C; and, therefore, CO and OD may 
be taken as symbols representing the raé of increase of the 
abscissa and ordinate of the curve, while CD will represent 
the rate of increase of the curve itself at the point C; and is at 
the same time tangent to it at that point. 


128 DIFFERENTIAL CALCULUS. 


So that if we designate the length of the curve by s, and 
consider CD as representing ds, we shall have CO=dx and. 
OD=dy for the point C of the curve. 

The tangent of the angle which the tangent line CD makes 


with the axis of abscissas is equal to aes and hence, calling 
this angle z, 
qo tang. v (1) 
and since CD» =CO” +0D’ we have 
ds* =ax* +ay* (2) 


We shall have frequent occasion to use these two equa- 
tions in investigating the properties of curves. 

(35) The usual method of obtaining equation (1) is, to sup- 
pose an actual increment BB’ (Fig. 16) given to the abscissa, 
and to find the corresponding increment C’O of the ordinate 
from the equation of the curve. The ratio of these two in- 


crements will not give the tangent of the angle v, which is 
DO 
CoO” 
and when they become infinitesimal or vanish, there is no 


; DO C/O 
difference between ao ada: 


This manner of reasoning we have already discussed in 
the introduction to this work, and its defects have been 
shown. We commit an error In giving an actual increment 
to the abscissa and ordinate, for their rates of increase are 
not obtained from any actual increase in value (except where 
the rate is uniform), but from the /aw of change derived 
from the equation of the curve; and the suppositive incre- 
ments which we give are not real, but symbolical, repre- 
senting what they would be, by the operation of the Zaw con- 


equal to but will approachit as the increments decrease, 


trolling them at that instant, and are, therefore, a symbolical 
expression of that law. 

The truth is that CO and OD, so far from being infinitely 
small may have any value whatever assigned to them; so 


THEORY OF CURVES. 129 


that we may consider them as varzadles whose simultaneous 
values always correspond to some point in the tangent line. 
In fact, if we differentiate the equation of a curve, and con- 
sider x and y as constants for the point of tangency, dx and 
dy may be considered as the variable coordinates of the tan- 
gent line, with the origin at the point of tangency, and the 
axes parallel to the primitive axes. Under these conditions 
the differential equation of the curve becomes the equation 
of its tangent line. This can easily be shown by a few 
examples. 

Ex.1. The equation of the circle with the origin at A 
(Fig 17) is 

y® =2Rx—x?* 
which being differentiated be- 
comes 

ydy =(R—x)dx 
If now we consider «x and y as 
constants for the point P, and 
dx(=PE) and a(=ET) as 
variables, we will replace the 
first by and x, and the latter 
by x and y, and we have Pig-o2: 

my=(R—n)x (1) 

which is the equation of the tangent line PT with the origin 
at P, while PE and TE are the abscissa and ordinate of the 
line, and # and z are coordinates of the new origin referred 
to the primitive one, or AB and BP. 

Suppose now we transfer the origin to O, the center of the 
circle. The formulas for transferring to a new origin ina 
system of parallel axes is 

y=bty’ and x=a+2' 
where a and & are the coordinates of the new origin. In 
this case a is equal to BO=AO—AB=R-—~z, and 4 is equal 
to PB=—wm, and hence by substitution, 


130 DIFFERENTIAL CALCULUS, 


m(—m+y’)=(R—x)(R—-n2+x’) (2) 
Calling x” and _y” the coordinates of the point of tangency 
for the new origin, we have 
x” =—R+n or R—-n=—2" 
also 
y —BP—y 
Substituting these values in equation (2) we have 
y"(=y" $y) =—0"(—2" +2) 
yy al x" Hy"2 + y"2®=RE 
or dropping the accents 
yy +a" =R? 
which is the equation of the tangent to the circle, the origin 
being at the center and x” and y’ the coordinates of the 
point of tangency. 
Ex. 2. If we differentiate the equation of the ellipse 


whence 


referred to its center and 35 
axes, we have ; 
A®’ydy + B*xdx=o ' 
Making | ae 0 
y(=BP) and x(=OB) Fig..28. 


(Fig. 18) constant and dy(=TE) and @(=PE) variables, 
and replacing y by y” and x by x", dx by x and dy by y, we have 
A®y"'y+B®x"x=0 
which is the equation of the tangent line to the ellipse, with 
the origin at P, the point of tangency, and the axes parallel 
to the primitive ones; x” and y” representing the coordinates 
of the new origin referred to the primitive one, and x and y 
the variable coordinates of the tangent line referred to the 

new origin. 
If we transfer the origin back to the primitive one we shall 
have 
x=a+x' and y=s+y’ 


THEORY OF CURVES. 131 


where a and # are the coordinates of the new origin — that 
is, the center of the ellipse. oo a=—x" and 6=—y" (for a 
is essentially positive while x” is essentially negative, and 3 
is essentially negative while y” is essentially positive), and 
substituting these values for x and y we have 
A®y"y’ + Bex" x ‘=Aty "24B2y"2=A2B3 
or dropping the accents 
A®y"y+ Bx x=A?B? 
£x. 3. Differentiating the equation of the parabola we 
have 
ply =pax 
Representing x, y, a and dy by x", y", xand y respectively, 
we have by substitutior 
YY = px 
for the equation of the tangent line to the parabola, with the 
origin at the point of tangency. If we transfer it to the 
vertex by making y=é-+y’ and «=a+.’, in which 6=—y" 
and a=—.x", we have 
—y" + yy" =px!’—px" 
in which x” and y” are the coordinates of the point of tan- 
gency for the origin at A. Hence y"?=24x", and substitut- 
ing we have aa & Sap 
— 2px! by y! =px'—px" 
or, dropping the accents, 
yy" =pxt px" =p(x+x") 
which is the equation of the tangent to the parabola at the 
point whose coordinates are x” and y", the origin being at the 
vertex. 
Ex. 4. Lastly we will take the equation of the Hyper- 
bola referred to its center and asymptotes 
aA A2 +B? 
Heme 
from which 
axdy +ya@x=0 


I 32 DIFFERENTIAL CALCULUS. 


Replacing y=PB (Fig. 19) by y’, x=AB by x’, dy=ET 
by y and zx=EP by x, we have . 
y LEX Y= 
in which x” and y” are the coordinates of the new origin 
referred to the primitive one, 
and x and y are the variable 
coordinates of the tangent line 
TP; the origin being at P and 
the coordinate axes, PM and 
PN, parallel to the assymp- 

totes. 

To transfer the origin to A 
the center of the hyperbola Fig. 19. 
make y=é-+y’, and «=a+x’; 6 being equal to —y” and 
- and substitute these values for x and y. This 


J 


Phew a 
Byres Y lf y / S\N wae 
i Ee ePaper Note 
or, dropping the accents 
MY Q g 
which is the equation of the tangent line to the hyperbola 
referred to its center and asymptotes. 

Thus it clearly appears that differentials are not infinitely 
small quantities, but are symdols to express the razs or laws 
of variation, which are, in fact, VARIABLE FUNCTIONS OF THE 
GIVEN VARIABLES. 


SIGN OF THE FIRST DIFFERENTIAL COEFFICIENT. 


(36) If x and y represent the coordinates of any curve, 
and while x increases uniformly y should have a positive 
value and, also, zzcrease, its differential will be positive and 
the curve will tend to Zave the axes of abscissas in a posi- 
tive direction; but if y should be decreasing while its value 


THEORY OF CURVES. 4 


is positive, its differential will be negative and the curve will 
approach the axis of abscissas on the positive side. 

Again if y has a negative value and zzcreasing, its differen- 
tial will be zega#ve, and the curve will be receding from the 
axis of abscissas on the negative side; while if it is decreas- 
ing (being still negative) its differential will be fosztive, and 
the curve will be approaching the axis of abscissas on the 
negative side (see definition of a differential, Art. 3). 

Hence the following rule: 

When the ordinate and tts first differential have the SAM¥. 
sign the curve ts receding from the axis of abscissas, and when 
they have DIFFERENT SIGNS ¢he curve 1s approaching that axis. 


Nore.— The differential of the independent variable is supposed to be constant and 
positive,and hence the sign of the differential coefictent is the same as that of the 
differential itself of the function. 


This rule may be illustrated by means of the circle (Fig. 
20) whose equation (the origin being at A) is 


y*® =2Rx—x? 
from which we obtain 


hae 
eet ee 
We see here that from A to C, y and its differential have 
the same sign, and the curve recedes | c 


from the axis of abscissas. Thesame 
is true of the curve from A to D. 
ieretcome (2 to: 4; and, from:D to B, 
where x is greater than R, the sign of 
y will be contrary to that of dy, and 
the curve approaches the axis of 
abscissas on both sides. ar 
We arrive at the same result if we Fig. 20. 


: ay . : 
consider - as representing the tangent of the angle which 


the tangent line makes with the axis of abscissas. From A 
to C and from A to D, where the curve leaves the axis of 


134 DIFFERENTIAL CALCULUS. 


abscissas, the sign of the tangent and of y are alike; while 


from C to R and from D to B their signs are contrary 
dy_R-«# 


oie becomes 


At the point A where y—a, the value of 


infinite, and the curve departs at right angles from the axis 
dy 
ax 


becomes zero, and the curve neither approaches nor recedes 
from the axis of abscissas. And this corresponds with the 
value of the tangent of the angle made by the tangent line 


of abscissas. While at the points C. and D the value 


with the axis of abscissas; at A and B, se oo and the angle 


is aright angle; at C and D, aa and the angle is zero; 


the tangent line being parallel to the axis of abscissas. 


—— 


oH € LONG Vil: 


DIFFERENTIALS OF TRANSCENDENTAL FUNCTIONS. 


PROPOSITION I. 


(37) To find the differential of a constant quantity raised 
to a power having a variable exponent. 
Let the constant quantity be represented by @ and the 
variable exponent by v,; then the function will be 
Qa 
If we add an increment to v which we will call m we shall 
have 


QetM=qrqgm (1) 
Differentiating this equation we have 
daaet*M™=qN"7qQ? (2 ) 


and dividing equation (2) by equation (1) we have 
agerm akg 
gotm — gd (3) 


This equation being true, irrespective of any particular 
value of w, it will be true for any value we may assign to it; 
hence the differential of a constant quantity raised to a 
power denoted by a variable exponent, divided by the power 
itself is a constant quantity, or 

da” 


| a° 
But we have seen (Art. 6) that the differential of the vari- 
135 


136 DIFFERENTIAL CALCULUS. 


able is always a factor in the differential of the function. 
Hence av will be a factor of C. Calling the other factor 
k we have 


C=ha7 
whence 

aa® a 

7? =kiv 
or 


da” =arkdv 
The problem now is to find the value of & For this pur- 
pose we expand a” by Maclaurin’s theorem and have 
a%—A-+ Bo+Cv* +Dz?.+ Lut + ete. 
in which 
da” a* ae? an” 
dy ~  2av® ~~ 2. 3% dys Tete. 
when v is made equal to zero. 
But we have found 


A=, B= 


Wary o7 
Woe Gok 
hence 
LOT) > sy to BB 
al a ) =aa k=a k* dv 
or 
ae 1 7 Be 
aye —° 
and similarly 
Pen ae eet a 
from which making v=o we have 
ve B3 pA 
A=1, B=kh, C= E pe ok 3° Lies a etc. 


° ° ss ° : 
Substituting these values in equation (1) we have 


ky? fyi = ptyt 


meee 2 eae 2. 3.04. 


° i ° 
and making v=; this becomes 


etc. . 


i 


TRANSCENDENTAL FUNCTIONS. 137 


1 


a®=1+1+4+2 ak a gat etC—2.715282-- 


If we represent this number by e we have 
1 


a*=e or axe 


If ¢ is made the base of a system of logarithms & would be 
the logarithm of a to that base. 

This was done by Napier, the inventor of logarithms, and 
the system having that base is called the Naperian system. 
We shall indicate the logarithms of that system by the nota- 
tion 4g., while the logarithms of other systems will be 
noted by Zog. We have, therefore, 

da* =a" log. adv 
that is, 

The differential of a quantity ratsed to a power denoted by a 
variable exponent, ts equal to the power multiplied by the Nafpe- 
rian logarithm of the constant quantity into the differential of 
the exponent. 


Proposition II. 


(38) To find the differential of the logarithm of a varia- 
ble quantity. 
Let the quantity be represented by 7 and its logarithm by 
v, the base of the system being represented by a. Thenwe 
have : 
r=@ anddr=arkdv 


whence 
adr I ae 
du= oR or @ Log. ey 
Representing = by JZ we have 
@ Log. r=M = = 


in which J7is the reciprocal of ‘te Naperian logarithm Pa 
the base a, and is called the modulus of the system of loga- 
rithms of which @ is the base. Hence 


138 DIFFERENTIAL CALCULUS. 


The differential of the logarithm of a variable quantity ts 
equal to the modulus of the system to which the logarithm be- 
longs, into the differential of the quantity divided by the quantity. 

In the Naperian system the modulus is, of course, ome. 


Hence in that system 


he 
a log. r=— 


from which we learn that in the Naperian system the rate of 
increase of the natural number, whatever may be its value, 
divided by the rate of increase of its logarithm, is always 
equal to the number itself. . 


Nortre.— This principle was used by Napier himself in constructing his table of 
logarithms, and explains his selection of his peculiar base. Hence he is one of the 
first discoverers of the Jrzzciple of the differential calculus, although he never applied 
it otherwise than to logarithms. 


(39) If we call e the base of the Naperian system of 
logarithms, athe base of any other system, 7 the logarithm 
of ~ to that base, z the Naperian logarithm of f, and s the 
Naperian logarithm of a@ we shall have 


pHa" pH=er a=e 
hence 
wherefore 


= pa Te S 
72— S71 OY 71— 3 amnte 


but 4 is the modulus of the system, and hence 


The logarithm of a number tn any system is equal to the Na- 
perian logarithm of that number multiplied by the modulus of 
the system. 

This property is not peculiar to the Naperian system. 
The logarithm of a number in azy system is equal to the 
logarithm of the same number in the common system mul- 
tiplied by the reciprocal of the common logarithm of the 
base of the new system. 

In fact, in any two different systems, the ratio between the 


TRANSCENDENTAL FUNCTIONS. 139 


logarithms of the same number is constant. Thus let @ and 
6 be the bases of two systems, # and z two numbers, and «+ 
and y their logarithms in the first, and z and v their loga- 
rithms in the second system; then 

m=ae n= m= n=l? 


whence 
ae =$" and a =o 
whence 
hed 2 
a=b® and a=bY 
whence 
. u_o 


or the ratio between the logarithms of the same number in 
two different systems is constant and equal to the ratio be- 
tween the logarithms of any other number taken in the same 
systems. Hence 

log. a: com. Log. a:: log. 10: com. log, 1o=1 
whence 


a FOG. G2 
com. Log. a= jog. 104 “og. @ 


as we have seen. 


Proposition IIT. 


(40) To find the differential of the sine of an arc. 

Let APD (Fig. 21) be a circle whose center is at O. Let 
POA be the given angle, then PB Y 
will be the sine of the arc AP, 
and also an ordinate of the circle 
to the axes OX and OY; while OB 
will be the cosine of the same 
angle, and also the abscissa of the 
point P of the curve, and AB the 
versed sine of the angle. 

From the equation of the circle 
with the origin at the center we 
obtain 


Fig. 21. 


140 DIFFERENTIAL CALCULUS. 


xax=—ydy 
and 
eae 
dx? =* - 
a 


If we represent the arc AP by s we have (Art. 34) 
ds* =ax* +dy* 


whence 
wee 2 2 
ds? => 2 met 1s ) yy2 
But 
x +y? =R? 
whence 
Ready? 
ads? = ms 
x 


from which we obtain 


x 
ay =e 
or 
; cos. 
SIDS R as (1) 
that is, 


The differential of the sine of an arc ts equal to the cosine of 
the arc tnto the differential of the arc divided by radius. 


PROPOSITION IV. 


(41) To find the differential of the cosine of an arc. 
From the equation 
COS. S=4/R?—sin.? 5 


we have 
—sin. s.d@sin. s 


/R*—sin.? s 
Substituting the value of @sin. s (Art. 40) and replacing the 
denominator by cos. s, we have 


i, COS Go 


TRANSCENDENTAL FUNCTIONS. I4I 


z —sin.scos.s.@s —sin. s (2) 
COS Sa ee 2 
IN GOSs-S R 


that is, 
The differential of the cosine of an arc ts equal to minus the 
sine of the arc into the differential of the arc, divided by radius. 


PROPOSITION V. 


(42) To find the differential of the tangent of an arc. 
From the equation 
' R sin. s 
tan? 797s ee 
cos. $ 
we obtain (Art. 14) 
R cos. s, @sin. s—R sin. s.d cos. § 
COSs2" S 
Substituting for dsin. s and dcos.s their values (Art. 40 


and 41) we have 


d@ tang. s= 


R cos.” s.as+R sin.® s.ds 
Plane cape eta 


R cos.* s 
or 
(cOS.£Scp sin. sj)ae oR? 
@ tang. s= cos.” $ an pery (3) 
that is 


The differential of the tangent of an are is equal to the square 
of the radius into the differential of the arc divided by the 
square of the cosine. 


PROPOSITION VI. 


(43) To find the differential of the cotangent of an arc. 
From the equation 


9 


~ 


Ot Gre or a 
tang. § 


we obtain 


—R*d tang. s’ 
COUR EE cee ey erat: 
tang.” § 


142 DIFFERENTIAL CALCULUS. 


Substituting for @tang.s its value from equation (3) we 
have a 28 
—R*ds ast 
Are 5 Cos.? 5 tang.2 5 sin.” Ase (4) 
that is : 
, Lhe differential of the cotangent of an arc is negative, and 
equal to the differential of the arc multiplied by the square of the 
radius, and divided by the square of the sine. 


Proposition: VII. 


(44) To find the differential of the secant of an are. 
From the equation 


R2 
sec. s=— 
COse's 
we have 
—R?dcos. s 
ES tf an meer ne 
cos.” § 


Substituting the value of @cos. s from equation (2) we have 


5 Eas sin, s 
SeC. S=O og , a (5) 


that is 

The differential of the secant of an arc ts egual to the differ- 
ential of the arc multiplied by the radius into the sine, divided by 
the square of the cosine. 


Proposition VIII. 


(45) To find the differential of the cosecant of an arc. 
From the equation 


R2 
coset. s=— = 
sin. ss 
we obtain 
—R?d/sin. s 
20S 


sin, *s 


TRANSCENDENTAL FUNCTIONS. 143 


Substituting the value of @ sin. s from equation (1) we have 


Cases 
2 COSCC. Si — 5 eG (6) 
Silt f 


that is 

The differential of the cosecant of an arc ts equal to minus the 
differential of the arc multiplied by radius into the cosine, divided 
by the square of the sine. 


PROPOSITION IX. 


(46) To find the differential of the versed sine of an arc. 
From the equation 


ver. sin. s=R—cos. s 
we have 
ad ver. sin. s=—d COS. $ 


Substituting for 7 cos. s its value from equation (2) we have 


f sin. § : 
a Verrsile s = R as (7) 


that is, 

The differential of the versed sine of an arc ts equal to the 
differential of the arc multiplied by the sine and divided by radius. 

(47) In these equations the arc is supposed to be the 
independent variable; and the generating point 1s supposed 
to flow around the circumference at a uniform rate. 

The differential of the arc may easily be found, consider- 
ing it as a dependent variable and either the sine, cosine, 
tangent, etc., as the independent one varying uniformly. 

If we take the sine as the independent variable we have 
from equation (1). 


as @ sin, $= ee ry sin. $ (8) 
cos. § V R*® —sin.? s 
If we take the cosine we have from (2) 
R 
i A COs. De etna ae COS. S. (9) 


144 DIFFERENTIAL CALCULUS. 


If we take the tangent we have from (3) 
2 2g 


COS ee, Pain 
as ae Shee tang. eaeod fet tang. s 
but 
sec.*s=R* -+tang.’s 
hence aoe 
tang.s 
Dae +tang.*s (10) 


Lastly, if we take the versed sine we have from equation (7) 


(h= @ Vet. sin. 


wacin es 
but since 
sin.ts = /(2R — ver. sin. 5) ver. sin. 5 
we have 
R d ver. sin. s 
as— (1 1) 


a/ (2R —ver. sin. s) ver. sin. s 


If, in equations (8), (9), (10) and (rr), we represent sin. s 
by z, cos. s by x, tang. s by y, and ver. sin. s by z, and con- 
sider R=1, we shall have 


ey ines, 7 (12) 
pens (13) 
fe (14) 
ee (15) 


From these equations we can find the rate of change in 
the arc when we know that of either of the four trigonomet- 
rical lines. 


SIGNIFICATION OF THE DIFFERENTIAL EQUATIONS OF THE 
TRIGONOMETRICAL LINES. 


(48) Describe the circle ADBE (Fig. 22) from the center 
O. Draw the diameter AB and the tangent TT’ at its 


TRANSCENDENTAL FUNCTIONS. 145 
extremity. Let sc, s’c’,s"c", and sc" be the sines of the arcs 
As, ADs’, ADBs” and ADBEs”, and OT and OT’ (negative 
for ADs’ and ADBs") be the secants of the same arcs; then 
AT will be the tangent of the arcs As and ADBs", and AT’ 
will be the tangent of the arcs ADs’ and ADBEs”’; ac, a’, 
oc’ and oc” will be the cosines of the same arcs, and Ac, Ac’, 
Ac" and Ac” will be their versed sines. 

Suppose now the generating point of the circle to move 
from A around through D, B and E back to A with a uni- 
form rate of motion, then 

From equation (1) we find that at the beginning where 
cos.=R the rate of increase of the sine is the same as that 
of the arc and is positive. As the sine increases the rate 
immediately begins to decrease, 
being always in proportion to the 
cosine, until the generating point 
arrives at D and the cosine be- 
comes zero. ‘The sine then ceases p 
to increase, being at a maximum, 
and its rate of increase is zero. 

In the second quadrant, the-sine 
although still positive, decreases, 
and hence its rate of change is 
negative as shown by the cos. s, 
which is negative in that quadrant, Fig. 22. 
at the point B the rate of decrease has become equal to the 
rate of increase of the arc, although the value of the sine 
itself has become zero, and hence at that point @ sin. s=—das. 
In the third quadrant the sine is negative and increasing, 
hence its rate is also negative, as is shown by the cosine 
which is negative. At the point E the negative increase 
ceases and the rate becomes zero as does the value of the 
cosine. In the fourth quadrant, while the sine is negative, 
it is diminishing, and hence its rate of change is positive 


146 DIFFERENTIAL CALCULUS. 


which is also indicated by the cosine which is positive for 
that quadrant. 

From equation (2) we learn that in the first and second 
quadrants where the sine is positive, the rate of change in 
the cosine is negative, and in the third and fourth quadrants 
where the sine is negative the rate change in the cosine is 
positive. We learn the same thing from the figure, for from 
A to B the cosine decreases, being positive, or increases, 
being negative; while from B to A, for the third and fourth 
quadrants, it decreases, being negative, or increases, being 
positive; the rate of change being at all times in direct pro- 
portion to the value of the sine. 

From equation (3) we learn that the rate of change in the 
tangent is at all times positive, and we learn the same thing 
from the figure, for as the arc increases from any point what- 
ever, the extremity of the secant which limits the tangent 
will move upward in a positive direction, and the tangent 
will increase in the first and third angles from A to positive 
infinity, and decrease in the second and fourth from nega- 
tive infinity to A; so that it will always increase positively 
or decrease negatively, and hence its rate of change is always 
positive (Art. 3). At A and B the rate is the same as that 
of the arc, and of the sine, being equal to as, while at D and 
E it is infinite. 

Similarly we learn from equation (4) that the rate of 
change in the cotangent is always negative, as it either 
decreases being positive, or increases being negative, for all 
points of the circle. 

From equation (s) we learn that the rate of change in the 
secant has the same sign as the sine, and hence is positive 
in the first and second quadrants, and negative in the third 
and fourth. By inspecting the figure we see that in the first 
quadrant the positive secant OT increases; in the second, 
the negative secant OT’ decreases; in the third, the nega- 


TRANSCENDENTAL FUNCTIONS. 147 


tive secant OT increases; and in the fourth, the positive 
secant OT’ decreases. At the point A the rate of increase 
of the secant is zero for sin. s=o, while the secant itself 
equals R (a minimum). We see also that the rate of the 
secant is equal to that of the tangent multiplied by the sine; 
and since the sine is (except at two points) always less 
than 1, the tangent increases faster than the secant until the 
arc equals go’, when the sine is 1 and the rates become 
equal being infinite. | 

Equation (6) will give a similar result for the cosecant in 
connection with the cosine and contangent. 

We learn from equation (7) that the rate of increase of 
the versed sine corresponds at all times to the value of the 
sine, and is therefore positive in the first and second quad- 
rants, and negative in the third and fourth. The figure 
shows that the versed sine increases positively from A to B, 
the rate of increase being an increasing one (corresponding 
to the sine) in the first quadrant, and then a decreasing one 
in the second, but still positive. At B it ceases to increase 
and begins to decrease, first at an increasing rate in the 
third quadrant, and then at a decreasing rate in the fourth, 
until at A the versed sine and the rate both become zero. 
Hence in the two last quadrants the rate is negative corres- 
ponding to the sine, while the versed sine is always positive. 
At A and B the rate of change is nothing, as it should be, 
since at those points the generating point of the circle tends 
to move ina direction perpendicular to the line on which the 
versed sine is laid off, and, therefore, does not tend to alter 
its value; while at D and E, the generating point moves 
parallel to the line of the versed sine, and, therefore, at those 
points they should have the same rate, and thus we find to 
be the case, for at D 

sin. § 


R 


We¥ere sm. s= Wsacas 


148 DIFFERENTIAL CALCULUS. 


and at E 


: sin. S 
Wer sites — R as =—ds 


because sin. s at that point is negative. 


VALUES OF TRIGONOMETRICAL LINES. 


(49) We are enabled by Maclaurin’s theorem to develop 
the sine and cosine of an arc in terms of the arc itself; for 
let s be the arc and z its sine, we shall have (Art. 24) 

w—=A+Bs+Cs?+Ds2-+4 etc. 


and (Art. 40, 41) making R=1 we have 


au au ; a> 4 
ei: =COS. S$, Wee SID. S, 7 a S7aCtC: 
making s=o we have ) 
Lu au au 
(zZ)==0, = (eae Se Ws Is ete. 
whence 
re 5 
w—=sIn, S=S—F— atita’ ty gic aA? CtG: 
If we represent by zw the cosine s then 
Mu ) au a*u- . 
Ws SID: 5) Te = C08. S$, 7g Sm. S, etc. 
Making s=o we have 
Lu \ au a*u 
(“)=1, Gee Te ets aya OE 
whence 
34 5# sf 
2#—=COS. Sl Ty aghig ag ee 5 eg 


These series are very converging, and for small arcs will 
give the length of the sine and cosine quite accurately. 
In order to apply these formulas, we take the length of a 


~ 


quadrant, which is iB the radius being 1; and this divided 


TRANSCENDENTAL FUNCTIONS. 149 


by go and then by 60, will give the length of one minute of 
arc, from which we can obtain the length of any number of 
minutes or degrees. Substituting the value of the arc thus 
found in the formulas, we obtain the length of the natural 
sine or cosine. 

If we wish these values for any other radius we shall have 
for sin. s=z 


Wit. COS, § “du Sines aru COS; S : 

—= Eee — ete 

as Re. as* Rieti? as? R38 
whence 


2 rn 


o a] 
ASIN § —5— Fe 7 ager a 5 5. RS ete. 
(50) We may in a similar way develop an arc in terms of 
its sine and cosine. 
Let s be an arc whose aah. is #, then (Art. 47) 


au / 1—u2’ du =u(1-1") : 
ons 5 
Tas =(1-4?)~ 2 —3u?(1-u") — 2 


making w=o , 
A=(z)=0, p=(@) =, c=1(—*) =o, p= 


Lu 
etc., hence 
wu ve 30° i 
Se be Ne 
2.3 2.4.5 
If we make w equal to the sine of 30° =4 we have for the 


value of the arc 


I 
SU 


s=arc whose sine is #=u+ 


3 ed 
Poe en ea uy 


the sum of which is 0.52359 nearly; and multiplying this by 
6 we have the length of the arc of a semi-circle, thus 


180° =x =3.14154 nearly 


which is also the approximate ratio of the diameter of a 
circle to its circumference. 


SOROMIB MERRICK L. 


OF TANGENT AND NORMAL LINES TO ALGEBRAIC 
CURVES. 


(51) We have seen (Art. 34) that when x and y represent 
the abscissa and ordinate of the curve, oy will represent the 


tangent of the angle made by the tangent line of the curve 
with the axis of abscissas. Now the equation of a line 
drawn through any given point is 
IV =ax—x') 
in which y’ and x’ are the coordinates of the given point, 
and athe tangent of the angle made by the line with the 
axis of abscissas. Hence for any curve in which ~’ and y’ 
are the coordinates of the point of tangency, the equation of 
the tangent line through that point will be 
Oe 
ED f =F le % ) 

The value of oY will, of course, be obtained from the 
equation of the curve, and by substituting that value we 
obtain the equation for the tangent line of that curve. 


EXAMPLES. 


£x.1. From the equation of the circle we have 
dy’ x’ 


a =——_— yl 


150 


TANGENT AND NORMAL LINES. I51I 


and hence 
yy = ae: =E)) 
Ws 
or 
yy +xx' =R? 


becomes the equation of the tangent line to a circle. 


Lx. 2. Inthe case of the parabola we have 


ay _p 
ax’ =e 
whence 
FP Pp , 
—y =a(x—-x 
IY a ) 
or 


yy =p(x+x") 


becomes the equation of the line tangent to a parabola. 


Ex. 3. The equation of the ellipse gives 


ay Bex 
dx Ay’ 
whence 
Rab. F 
B dow 2 = TAns at) 
or 


A® yy’ +B? xx’ =A*B? 
becomes the equation of the line tangent to the ellipse. 


fx. 4. From the equation of the hyperbola referred to 
its center and asymptotes, we have 


Ti ores Je 
ax x’ 
whence 
DAS ' 
Vis reat) 
or 


A2 + B2 
2 


yx xy’ = 


152 DIFFERENTIAL CALCULUS. 


becomes the equation of the line tangent to the hyperbola, 
referred to its center and asymptotes as coordinate axes — 
as in Art. 35. 

Since the normal line is perpendicular to the tangent, if 
a represent the tangent of z¢s angle with the axis of abscis- 
sas, then it will be equal to a where @ represents the tan- 
gent of the angle of inclination of the zamgen¢ line. Hence 


and substituting this value in the equation 

IY =a (x—2') 
we have 

ax 
—y =——(x—-2’ 
afeas ays ) 

for the general equation for the normal line, and it may be 
found for any particular curve by obtaining the value of 
—F from the equation of the curve, and making the sub- 


stitution as in the case of a tangent line. 
PROPOSITION I. 


(52) To find the general expression for the length of the 
subtangent to any curve. 

Let AP (Fig. 23) be any curve of which PT is the tangent 
at the point P, TB the subtangent, 
PN the normal, and PB the ordinate; 


then from the triangle TPB we have ; 
PB=TB. tang. PTB 
whence = + 
= A B N 
TB ~ tang. PTB Fig. 23. 
but 


ay 
tang. PTB Bear and PB=y 


TANGENT AND NORMAL LINES, 153 


hence 


that is, 
The subtangent to any curve 1s equal to the ordinate into the 
differential of the abscissa divided by the differential of the 


ordinate. 
Proposition II. 
(53) To find the general expression for the length of the 


tangent to a curve. 
From the triangle PTB (Fig. 23) we have 


fo pik ax* 
PT= 2 : pereies 
=/9 ae ay ay” Ve T dy” 
that as; 


The length of the tangent to any curve ts equal to the ordinate 
into the square root of one plus the square of the differential 
coefficient of the abscissa. 


whence 


Nore.— By the ‘‘ length of the tangent” is meant that part of the tangent line 
between the point where it intersects the axis of abscissas and the point of tangency on 
the curve. 


Proposition III. 


(54) To find the length of the subnormal to any curve. 
Since the triangle PBN (Fig. 23) is similar to the triangle 
PBT, we have the angle BPN=BTP, and hence 


BN=PB. tang. BPN 


or 
ay 


BN= me 


that is, 


154 DIFFERENTIAL CALCULUS. 


The subnormal ts equal to the ordinate into the differential 
coefficient of the ordinate. 


PROPOSITION IV. 


(55) To find the length of the normal to any curve. 
Since py” (Fig. 23) is equal to pp? +BN’, we have 


$ dy? dy” 
PN=A/ 9° 49° a =IN/ 14 oe 
that is, 


The length of the normal line ts equal to the ordinate into the 
square root of one plus the square of the differential coefficient 
of the ordinate. 


Note.— By the “‘ length of the normal” is meant that part of it which lies be- 
tween the point of its intersection with the axis of abscissas and the point of the curve 
to which it is drawn, 

(56) The following examples will show the application of 
these formulas to particular cases. 

£x. 1. From the equation of the circle we have 

ax y 


; ay oe 
hence the subtangent (Fig. 24) is 
ax we PB 


Mire ar et 10) 


a result that we also obtain 
from geometry. P 


Age. vilhevienoth *onetveecy. 


tangent to the circle is AB 0 
Fig. 24. 


We have also by geometry 
TR roa 


TANGENT AND NORMAL LINES, 155 


whence 


£x. 3. The normal line of the circle is 
. oP aime 
PO=y\/1 $y 1 eyes 


Ex. 4. The subnormal of the circle is 


BO 
J dx Gg Pe 
Ex.5. From the equation of the parabola we have 
ave 
dy ?p 
hence the subtangent (Fig. 25) is 
Len OE SEs 
AE mm) gene ape —2x*—2A5 
fae ae 
a result which we have also 
from geometry. ° 
fix. 6. The tangent of che ee: 
parabola is Ae B N 


Fig. 25. 
au? ie y* Lace Sees 
TRay\/ te IN 1+ =N/ 9? tor HV Fa 


We have just seen (Ex. 5) that TB=2AB, hence 
7B” =4aB* =4x? 


whence 
TP=V y?+4x?=V pp? +R? 
as is evident from the figure. 


Ex.7. The subnormal to the parabola is 


YY p 
SOS eas gee roy! 


as we find from geometry. 


156 DIFFERENTIAL CALCULUS. 


fx. 8. The normal to the parabola is 


ady* ine Sead 
PN =y\/'x SSN eae a +2?=V ve +5y° 
which is evident from the figure. 


£x.9. From the equation of the ellipse we have 
ay Bix 
Ea gis 
and the subtangent (Fig. 26) is 
Be ial aNd ft Bk Nhe 
ays Bx x 
This value for the subtangent 
does not contain B, and hence P 
is the same for all ellipses hav- 
ing the same major axis, the 
abscissa being the same. Hence 0 B T 
the tangent to the circle at P Fig. 26, 
will intersect the axis of abscissas at T, and 
PB OP —OB  A?—<? 
ucteipere visite 
as we have already found, 


BT=y 


SH Grit O:N=s Neb ieie 


DIFFERENTIALS OF CURVES. 


(57) We have seen (Art. 34) that the differential of a 
curve is equal to the square root of the sum of the squares 
of the differentials of the ordinate and abscissa. Hence to 
find the differential of any particular curve, we must find 
from its equation, the differential of one of the coordinates 
in terms of the other. The formula will then give the dif- 
ferential of the curve in terms of a single variable. 


EXAMPLES. 


£x.1. From the equation of the circle we have 


: LAX LIX 
Sie more a el 
y V R2—x? 
hence if we designate the arc by w we have 
Soe ae Rex 
MU=V/ det +d =A / dec? Sy ar Pe = 
+4 a Rete e V R?2—x? 


which is the differential of the arc of a circle. in terms of 
the variable abscissa. 


Ex. 2. From the equation of the parabola we have 


dx= Fay 
and calling the length of the arc z we have 
St ot DE | SY corer 
au=NV dx® tay? =N/ dy? +—— == V pty? 


157 


158 DIFFERENTIAL CALCULUS. 


TPES OA ELOnT ns equation of the ellipse we have 


3 oe 
We =F (ata? Jand g=— AP 
. hence 
A eat : Bix? dx? Bex dx? 
re Vo pe ase ee 
spate ag As Bi (atx?) AlAs) 
A2 
hence 
—B?)x 
=V a xe? + dy’ +dy* =f /=— ca 


DIFFERENTIALS OF PLANE SURFACES. 


(58) Every surface may be considered as generated by 
the flowing of a line. 

If we wish to obtain the raze at which the surface is gen- 
erated we must, if possible, consider every point in the line 
to be moving in a direction perpendicular to the line itself, 
if it is straight, or to its tangent at that point if it is a curve. 
For the only method of estimating the rate at which the 
surface is generated is by means of the length of the gener- 
ating line and the rate with whichit moves. Now unless the 
movement is made in a direction perpendicular to the line, 
the rate of zés motion will be no criterion of the rate with 
which the surface is generated. Thus the line AB (Fig. 27) 
moving in a direction perpendicular to itself will generate 
the rectangle ABCD, but if ut 0 


C 
move at the same rate in any se 
other direction as Ad, the surface | 
generated in the same time will 


be less until if it should move 
in its own direction it would . Fig. 27. 

generate no surface whatever. Hence in order that 
the szmple movement of the line may be _ properly 
an element in estimating the rate of generation of the 
surface, 1t must always be supposed to take place in a 


_—- * 


DIFFERENTIALS OF CURVES. 159 


direction perpendicular to the line itself at every point. 
Otherwise we must include in our estimate of the rate, the 
sine of the angle made by the line withthe direction in 
which it moves, which in most cases would be inconvenient, 
and, in many, impracticable. 

(59) A plane surface may be generated in two ways by a 
straight line —by moving so as to be always parallel to 
itself, or by revolving abouta fixed point. If it is supposed 
to be generated by the first method, and the boundary line 
is symmetrical about the axis of abscissas, the ordinate of 
the line is taken as the generatrix, and while it moves par- 
allel to itself one of the extremities is in the line, and the 
other in the axis, and thus half the surface is generated. 

Thus if we consider AB, the diameter of the circle ADB 
(Fig. 28) as the axis of abscissas, we would consider the 


upper half of the circle as generated by De 
the ordinate DE moving parallel to it- i 
self, one extremity being alwaysin the ,/_|E , 


curve and the other in the axis AB. 

And similarly with-a surface bounded 

by any other line that is symmetrical ame 
about the axis of abscissas. 

(60) The differential or rate of increase of any surface 
at the moment the generating line has arrived at any given 
position, such as BC (Fig. 29), will be represented by the 
increment that would take place (Art. 2) in 
a unit of time if the surface should in- 
crease uniformly, after the generating line | 
should leave the position BC at the same 4 —p 4 
rate as that with which it arrived there. Fig. 29 
Now, in order that the increment may be uniform, the gen- 
erating line must maintain the same length and flow at an 
unvarying rate. Thus let AC be the curve and CB the gen- 
erating line of the surface ACB; and let Bd represent the 


160 DIFFERENTIAL CALCULUS. 


uniform increment of AB in a unit of time, at the same rate 
as at B; then the rectangle CcdB will represent what zwould/ 
ée the uniform increment of the surface during the same 
unit of time at the rate at which it was increasing at CB. 
And, hence, if we consider the increment Béas the sym- 
bol representing the rate of increase of AB, the rectangle 
CcéB will be the proper symbol to represent the rate of in- 
crease or differential of the surface ACB. But the rectan- 
gle is equal to BC. Bd ; and if we call AB x, BC will be y, 
and Bé the differential of x. Hence CcdB, or the differential 
of the surface will be 


yax 
that is 


The differential of a plane surface bounded by the axis of 
abscissas and a curve, ts equal to the ordinate multiplied by the 
differential of the abscissa. 

(61) In order to obtain the differential of any particular 
plane surface we must know the equation of the line that 
bounds it, in order that we may eliminate x or y from the 
formula. We shall then have the differential of the surface 
in terms of a single variable. 


EXAMPLE I. 


6 


(62) To find the differential of a triangle. 
Let ABC (Fig. 30) be the triangle, referred to A as the 


originand AB and AD as coordinate axes. G 
The equation of the line AC is : ) 
y=ax 
hence yi 
yax=axdx | B 
which is the differential of the surface of Bias: 


the triangle; a being the tangent of the angle made by the 
line AC with the axis AB. 


DIFFERENTIALS OF CURVES. 161 


EXAMPLE 2, 


(63) To find the differential of the surface of a semi- 
circle. | | 
If we take the equation of the circle with the origin at 
the extremity of the diameter, we have 
V=HV 2Rxe—x? 
whence 
VEX =AKN 2Re—x? 
which is the differential of the surface of a semicircle, the 
origin being at the extremity of the diameter. If we take 
the origin at the center we have 
yaAx=AXV/ R2— 4x2 


EXAMPLE 3. 


(64) To find the differential of the surface of a semi- 
ellipse. 

If the ellipse be referred to its center and axes, we have 
from its equation 


hence 
B re obs 
yadx =A eK A? —x? 


which is the differential of the surface of the semi-ellipse. 


EXAMPLE 4. 


(65) To find the differential of the surface of a semi- 
parabola. 

From the equation of the parabola referred to its vertex 
and axis we have 


J=V 2px 


162 DIFFERENTIAL CALCULUS. 


hence 
ae 
Y@X=AXN/ opx=V/ 2p x" aX 
which is the differential of the surface of a semi-parabola. 


DIFFERENTIALS OF SURFACES OF REVOLUTION. 


(66) A surface of revolution is one which may be gener- 
ated by a curve revolving about a line in the same plane. 
Every point in the revolving curve will describe a circle 
whose plane is perpendicular to the axis of revolution and 
whose center is in the axis. Any plane passed through the 
axis will cut from the surface a curve which is identical 
with the revolving curve. 

Such a surface may also be supposed to be generated by 
the circumference of a circular section, made by a plane 
passed through the surface perpendicular to the axis, mov- 
ing parallel to itself with its center in the axis of revolution, 
and its radius varying in such a manner, that its circumfer- 
ence shall always intersect the meridian section or directing 
curve. 

The rate of increase, or differential, will be determined, as 
in other cases, by finding the surface that qwould be generated 
in a unit of time, if the generating circle were to move dur- 
ing that time, without change of magnitude at a uniform 
rate, equal to that with which it arrived at the point of dif- 
ferentiation. Such a surface would be equal to the circum- 
ference of the generating circle into the line which repre- 
sents its rate of motion. Now the center of the generating 
circle is supposed to move along the axis at a uniform rate, 
hence its circumference will move along the directing curve 
at the same rate as the generating point of the curve; so 
that the line which represents this rate will be the same as 
the differential of the curve. 

Moreover the suppositive differential surface that we are 


DIFFERENTIALS OF CURVES, 163 


seeking must be generated at a uniform rate, and hence the 
diameter of the generating circle must not change; so that 
the surface will be that of a cylinder, whose base is the cir- 
cumference of the generating circle at the point of differen- 
tiation, and its height, the line which represents the differ- 
ential of the directing curve at the same point. 

If now we take the axis of abscissas as the axis of revo- 
lution, the radius of the generating circle will be an ordinate 
of the directing curve and the differential of the curve will 
be VW @x® +dy? (Art. 34); and hence calling the surface of 
revolution S, we have 

IS=27yV/ de® + ady® 
that is 

The differential of a surface of revolution ts equal to the cir- 
cumference of the generating circle tnto the differential of the 
directing curve. 

To apply this formula we obtain from the equation of the 
directing curve, the value of one variable in terms of the 
other, and by substitution obtain the differential in terms of 
a single independent variable. 


EXAMPLE 1I.., 


(67) To find the differential of the surface of a cone. 
In this case the revolving line is straight, and not a curve, 
but the principles of the rule apply 
equally well. C 
Let AC (Fig. 31) be the revolving ele- 
ment of the cone, and AB the axis of A 
revolution and of abgcissas, the origin B 
beingat A. Then we have for the equa- Fig. 31. 
tion of the line AC, y=ax and dy=adx, a being the tangent of 
the angle BAC. 
Substituting these values in the formula we have 


AS=27AKV Q? dx? 4 de? =27AXAKY Q? 44 


164 DIFFERENTIAL CALCULUS. 


EXAMPLE 2. 


(68) To find the differential of the surface of a sphere. 
From the equation of the circle we have 


r XxAX d dy? eS 2 dx? 
Ly — — samara) —_ 
oy ae ean on te 
hence 
x” 9 
2myV dx® + dy® =an eee 
y pW = 
whence 


@S5=27Rax 
for the differential of the surface of a sphere. 
As the entire expression besides dx is composed of con- 
stants, we infer that the surface of a sphere increases at the 
same rate as the axis. 


EXAMPLE 3. 


(69) To find the differential of the surface of a parabo- 
loid of revolution. 

From the equation of the parabola we have 
yay” 

2 


lh e 
ee a and dx? = 


hence 


AS=27yV/ x2 +dy? — zen ay 


EXAMPLE 4. 


(70) To find the differential of the surface of an ellipsoid 
of revolution. 
We found (Art. 57) that the differential of the elliptic 
curve 1s 
At— iS by 
a2 ae \/ eee G AGEs (ABH 


hence if we substitute this expression in place of V dx? +dy? 


DIFFERENTIALS OF- CURVES. 165 


in the formula, and for y its value derived from the equation 
of the ellipse, we have 


aS =arh/ 2; (At 2?) 232). de hoe pest B! 


which becomes by reduction 


as 


x V At—(A?—B?)x? 


DIFFERENTIALS OF SOLIDS OF REVOLUTION. 


(71) A solid of revolution is one which is described or 
generated ‘by a surface, bounded by a line and the axis 
about which it revolves. If this axis be that of abscissas, 
then the ordinates of the bounding line will describe circles, 
of which they will be the radii and the centers will be in 
the axis. Any one of these circles may be considered as 
the generatrix, which describes the solid by moving parallel 
to itself, as in the last case. Butitis now the surface of the 
circle and not merely its circumference that generates ; and 
its movement is measured along the axis, the rate being the 
same as that by which the abscissa of the directing curve is 
increasing. ; 

Now the rate of increase of a solid of revolution is meas- 
ured by asuppositive increment that qwould be described in a 
unit of time, by the generating circle moving uniformly 
along the axis, with its diameter unchanged at the same 
rate as that with which the abscissa is generated. Hence 
such a solid would be a cylinder whose base is the genera- 
ting circle, and whose altitude is the line representing the 
differential of the abscissa. But the area of the generating 
circle is zy”, and the altitude of the cylinder is dx ; hence 
the cylinder representing the differential of a solid of revo- 
lution, would be expressed by the function, 

my” atx 


166 DIFFERENTIAL CALCULUS. 


hence, 

The differential of a solid of revolution ts equal to the gener- 
ating circle multiplied by the differential of the abscissa of the 
bounding line. 


EXAMPLE I. 


(72) To find the differential of the volume of a cone. 

If we take the vertex of the cone for the origin, and the 
axis of abscissas for its axis, the equation of the revolving 
line will be 

| yHax 
and hence calling v the volume of the cone we have 
dyv=ry*dx=Ta*x* dx 
in which a is the tangent of the angle made by the revolv- 
ing line with the axis, and x the distance from the vertex to 
the base of the cone. 


EXAMPLE 2. 


(73) To find the differential of the volume of a sphere. 

If we take the origin at the extremity of the diameter, the 
equation of the revolving semi-circle will be 

y® =2Rx—x? 
in which Ris the radius of the sphere, and x any portion of 
the axis of revolution measured from its extremity at the 
origin until it equals 2R; hence the formula for the differ- 
ential becomes 
du=ry*dx=n7(2Rx—x? )dx 


EXAMPLE 3. 


(74) To find the differential of the volume of an ellipsoid 
of revolution. 
If we suppose the semi-ellipse to revolve about its major 


DIFFERENTIALS OF CURVES. 167 


axis, it will generate an oblong ellipsoid of revolution, oth- 
erwise called a prolate spheroid. If we take the origin at 
the extremity of the transverse axis, the-equation of the 
ellipse is 


and hence the formula for the differential of the volume 
becomes 


9 


B* 
du= ny dx =n (2Aa—x* dec 


in which A is the semi-transverse and B the semi-conjugate 
axis of the ellipse which generates the ellipsoid of revolu- 
tion. 

If we take the conjugate axis of the ellipse for the axis 
of revolution and its extremity for the origin, we have 


9 


A* 
=p2(eBx—x*) 


° 
~ 


y 
and 


adv =753(2Ba—x \dx 


In this case the volume is an oblate ellipsoid, or other- 
wise, an oblate spheroid. 


IX AMPLE 5. 


(75) To find the differential of the volume of. a parabo- 
loid of revolution. 
The axis of the parabola being the axis of revolution, and 
the origin at the vertex, we have 
dy=ry* dx =27pxdx 
in which / is the parameter of the revolving parabola that 
generates the volume. 


Ss OL OM bs ORs i.e 


POLAR CU RIGS: 


PROPOSITION I. 


(76) To find the tangent of the angle which the tangent 
line makes with the radius vector. 

Let CC’ (Fig. 32) be any curve of which we have the 
polar equation. Let P be the pole; PM=-; the radius vec- 
tor; Pb=R, the radius of the CeO 
measuring arc dd; bPd, the vari- 
able angle=v, and OT, the tan- 
gent to the curve at the point 
M. Produce the radius vector, 
PM to R, draw RO perpendic- 
ular to PR, meeting the tangent 
in O; draw ON parallel to RM, 
and MN parallel to RO, meet- 
ing each other in N. Join PN, 
and draw ad parallel to MN meeting PN in a. Suppose the 
radius vector to revolve around the point P in the direction 
from @ towards &. . | 

The generating point, being supposed to have arrived at 
the point M of the curve will be subject to two distinct, 
although mutually dependent, laws or tendencies. One of 
these tendencies arises from the law of change in the length 

168 


Fig. 32. 


POLAR CURVES. 169 


of the radius vector, which causes the generating point to 
move outward in the direction of its length. The other ten- 
dency arises from the revolving motion of the radius vector 
which causes every point in it (including, of course, the 
generating point) to move in a direction perpendicular to 
itself. Hence in this case, the law would incline the gen- 
erating point to move in the direction MN. If then we 
take the distance MR to represent the uniform outward 
movement that would take place, under the influence of the 
first law in a unit of time, it will represent the rate of 
change in the radius vector arising from that law, and is, 
therefore, the symbol of that rate; that is 
MR=ar 
If we take MN to represent what would be the uniform 
movement under the second law in the same length of time, 
it will represent the rate with which it tends to move in the 
direction MN arising from ¢#a¢ law. Now as both these 
laws act together without disturbing each other, the gener- 
ating point, if left to its tendency at the point M would move 
in such a direction as to obey both laws or influences at the 
same time; and hence at the end of the same unit of time 
would be found at O, having described the line MO; the 
departure from the line MN being ON=MR, and the depar- 
ture from the line MR being RO=MN. But the generating 
point of a curve, if left to its tendency at any time would 
move in a line tangent to the curve, and since the line MO 
would be uniformly described ina unit of time, it represents 
the rate of increase of the curve, and is also tangent to it. 
Hence if we call the length of the curve w we have 
MO=du 

The point 4 at the intersection of the radius vector with 
the arc of the measuring circle, Zends fo move in the direction 
ba, and if left to that tendency would describe that line in 
the same time that the generating point would describe the 


170 DIFFERENTIAL CALCULUS. 


line MN;; for the rate of movement of 2 is to that of M as 
Pé is to PM, or as dais to MN. If then we consider J-as a 
point in the arc of the measuring circle, we inay consider 
ba as representing its rate of increase, that is the rate of in- 
crease of the angle JPd, and hence 
ab=av 
But from the triangles PMN and Péa we have 
PZ: PM::a6: MN 


Eee 


hence 
PM .ab_rdv 
Ny Se RL 
Now the tangent of the angle PMT=MON is equal to 
a MN SN 
NO SeaM 


and substituting the value of MN just found and of MR, 
we have 
rav 
Tang. PMT ere 
that is 
The tangent of the angle which the line tangent toa polar 
curve makes with the radius vector 1s equal to ve radius vector 
into the differential of the measuring angle divided by that of the 
radius vector. 
(77) Since MNO (Fig. ae is a Ben angled triangle, we 
have 


MO =MN 4NO =MN 4+MR 


hence by substitution 
2 


: r* ay” 
du? = R2 


+dr? - 
whence 
MazV Peder Rare 


or making R=1 
LU=V 72 dy® + dr? 


POLAR CURVES. by 


that is 

The differential of the arc of a polar curve ts equal to the 
square root of the sum of the squares of the-radtius vector into 
the differential of the measuring angle, and of the differential 
of the radius vector. 


Proposition II. 


(78) To find the subtangent of a polar curve. 

The subtangent of a polar curve is the projection of the 
tangent on a line drawn through the 
pole perpendicular to the radius vector 
of the point of tangency. 

Hence if PT (Fig. 33) be drawn per- 
pendicular to PM, meeting in T the 
tangent to the curve at the point M, 
then PT will be the subtangent. Since 

ee ME tang Mer 
iiss R 
we have by substitution 


spells MEE 
Rear ae te 


that is 
The subtangent of a polar curve ts equal to the square of the 
radius vector tnto the differential of the measuring arc divided by 
R into the differential of the radius vector. If we make R=r 
rai 


we have PT =~ =tangent of PMT. 


Proposition III. 


(79) To find the value of the tangent to a polar curve 

The tangent to a polar curve is that part of the tangent 
line which lies between its intersection with the subtangent 
and the point of tangency. 

Hence MT (Fig. 33) will represent the tangent, and since 


172 DIFFERENTIAL CALCULUS. 


+8 Go eee 
MT =PM + PT 
we have 
—? Fi dct 
=r? +—_—_. 
MT =r TRi7,2 
or 
vr? dv r Se i Decl tt 
nes phhst ag A Ca ga bier pee 
MT=rA/ 1+ R373 = Rap’ Rear? +r*do 
or making R=1 
r* dy" 
MT=rA/1 Bee 


PROPOSITION IV. 


(80) To find the subnormal to a polar curve. 

The subnormal of a polar curve is the projection of the 
normal line on a line drawn through the pole perpendicular 
to the radius vector for that point of the curve to which the 
normal is drawn. 

Hence if MB (Fig. 33) be a normal at the point M, BP 
will be the subnormal. 

The triangles MBP and MTP being similar, the angles 
MBP and PMT are equal, and since 


tang. MBP 
Pe NMe be R 
we have 
rav 
r—BP a 
or 
z= Rdr 
BR= ae 
that is 


The subnormal of a polar curve ts equal to radius into the aif- 
ferential of the radius vector, divided by the differential of the 
MeCASUYING ALC. 

(81) The normal line MB (Fig. 33) is equal to 


V ue? +PB” 


— 


POLAR CURVES. 173 


or 


eRe 
MB=A/7? +—— 


adv" 

(82) While the point at the extremity of the radius vec- 
tor describes the line of a polar curve, the radius vector 
itself generates the surface bounded by the curve. 

Now the point M of the line PM (Fig. 32) tends to move in 
the direction MO, and every other point in the line PM will 
tend to move in a direction parallel to MO, and ata rate 
proportional to its distance from the fixed point P. Hence 
if the point M were to be found at O, the line PM would 
assume the position PO, and the triangle PMO would be 
that which would be generated at a uniform rate by the 
radius vector PM if left to its tendency when in that posi- 
tion, so that the triangle PMO is the true symbol to repre- 
sent the rate at which the surface bounded by the polar 
curve is generated, or, designating the surface by O, we have 


triangle PMO=dO 


But since ON is parallel to MP, we have 


triangle PMO=triangle PMN 
and 
PM x MN 


PMN= 
and substituting here the values already found for these 
terms, we have 
r* do 
aly 


@aOo= 

hence 
The differential of a surface bounded by a polar curve ts 
equal to the sguare of the radius vector into the differential of 
the measuring arc divided by twice tts radius. 


SPIRALS, 


(83) If a right line revolve uniformly in the same plane 
about one of its points, and a second point should at the 


174 DIFFERENTIAL CALCULUS, 


same time approach to, or receded from the fixed point, 
according to some prescribed law, it would generate a curve 
called a spiral. 

The fixed point is called the pole, and the curve generated 
during one revolution of the line is called a sfzve. There 
being no limit to the number of revolutions of the line, the 
numberof spires 1s infinite, and a line, drawn through the 
pole will intersect the curve in an infinite number of points. 

Hence there can be no algebraic relation between the 
ordinates and abscissas of the curve, and its conditions 
must be expressed by a polar equation which will be in the 
form 

r=F(v) 
in which ~ is the radius vector and v the measuring arc of 
the variable angle. 


SPIRAL OF ARCHIMEDES 


(84) This spiral is one in which the radius vector is con- 
stantly proportional to the corresponding arc which measures 
its angular movement. - Hence its equation will be 

r=av (1) 

The curve may be constructed in the following manner. 
Divide the circumference of 
the measuring circle into eight 
equal parts by the radu AB, 
AC, AD, AE, etc., (Fig. 34); 
alsorthe radius A.B aintosthe 
same number of parts. Then 
lay off from the center one of 
these parts on AC, twoon AD, 
three on AE, and so on, there 
being eight on AB, nine on 
AC, ten on AD, and so on. Fig. 34. . 
Through the points thus found draw the curve commencing 


a 


POLAR CURVES. 175 


at the pole. The radius vector will be to the corresponding 
measuring arc as the radius of the measuring circle is to the 


: I 
circumference; or @ will be equal to cag 


Norter.— In this construction we have supposed the radius of the measuring circle to 
be equal to the radius vector after one revolution. Of course any other proportion 
might be taken, but as the magnitude of the spiral does not depend on that of the 
measuring circle, the radius of the latter may always be taken equal to the radius vec- 
tor after one revolution. 


If we differentiate equation (1) we have 
adr =adv (2) 
In a polar curve (Art. 76) the tangent of the angle which 
the tangent line makes with the radius vector is equal to 
av 
“ar 


and from equation (2) we have 


hence the tangent of the angle APT is equal to <, or in this 
case, to 
2mr 
This tangent will after one revolution be equal to 
a7R 
(85) The subtangent of a polar curve (Art. 78) is 
vr? dv | 
Rar 
which becomes for this curve 
A gh 
Ra R 
or, making y=R, we have 
subtangent=27R 
equal to the tangent of the angle made by the tangent line 
with the radius vector; and also equal to the circumference 
of the circle described by the radius vector as a radius, 


176 DIFFERENTIAL CALCULUS. 


when the point of tangency is at the circumference of the 
measuring circle. 

If we make v=za2zR, that is, if the tangent be drawn to 
the curve after z revolutions of the radius vector, then 


r r 
a 
v neorR 


whence 
dv 1 n2nR r°dv 
it a and Ray 7277 
that is 
After revolutions of the radius vector, the subtangent 
is equal to z times the circumference of a circle described 
by the radius vector as a radius. 


For the subnormal whose value is (Art. 80) 


Rar 
adv 
we have 
TENG MEST, 
do “Te 
hence 


Rr 
subnormal as 


If the normal is drawn at the point B then 
v=27R 
and we have 


4 
subnormal——— 
Dit 


that is 
The subnormal is equal to the radius of a circle of which 
r=R is the circumference. 


THE HYPERBOLIC SPIRAL. 


(86) The equation of the Hyperbolic Spiral is 
rv=ab 


POLAR CURVES. 177 


in which 7 is the radius vector, v the measuring arc, a the 
radius of the measuring circle and 4 the unit of the meas- 
uring arc —aé being, of course, a constant quantity. It is 
called a Hyperbolic spiral because its equation resembles that 
of a hyperbola referred to its center and asymptotes. 

To construct this curve describe a circle with a radius 
PA (Fig. 35) C N 
equal to a. 
Lay off from 
aT eer atC 
AB=das the © 
unit of the 
measuring 
ALGw Oy aanc 7 
continue this Fig. 35. 
division around the circumference of the measuring circle. 
Also lay off As=$6, Ar= 7d, and so on. 

Through these points of division in the circumference 
draw the radii PB, PC, PD, PE, and so on, and produce the 
radii Ps and Pv. On these radu lay off Pe=ta, Pd=ta, 
Pe=ta, and soon; alsoPO=2a, PQ=4a. Draw the curve 
through the points thus found. 

The radius vector multiphed by the measuring arc, count- 
ing from A, will always be equal to the radius of the meas- 
uring circle into the unit of the arc, that is 

rvu=ab 

We see from the equation that ~ increases as v diminishes, 
and wie versa. If v=o r becomes infinite, and hence the 
radius vector, through A, will never reach the beginning of 
the curve. If ~=o then v will be infinite, hence the curve 
will never reach the pole. 

If we take any point O in the spiral and join OP, then 
OP will be equal to, and the arc As=v. Draw OR per- 
pendicular to PA and we have 


178 DIFFERENTIAL CALCULUS. 


—— 


OP.sin.As y*sin. zg 


OR 
PA a 
hence 
COR Va7 
Y — Ee 1 ma ae 
sin, v 
and since | 
rvu=ab | 
we have | 
OR. av 
SEE he | 
sin. v 
whence 
sin. v 
CMe Ss :, b 


As sin. v is always less than z, the line OR will always be 
less than 4, but may be made to approach that value as near 
as we please. Hence if we draw a line MN parallel to PA, 
at a distance from it equal to d=arc AB, it will be an asymp- 
tote of the curve. 


Since 
CL 
arr 
the subtangent (Art. 78) 
FIQD IA PH 
Rar a@ 
and since 
rv=ab 
we have 
subtangent = —4=—arc AB | 


a constant quantity. Thus Pm or Pa=AB or PM | 
Also the tangent of the angle made by the tangent line 

with the radius vector (Art. 76). 

rau 

ar 


=—v 
thatvs; 

The tangent of the angle which the tangent ine makes with the 
radius vector is negative and equal to the arc which measures the 
angle made by the radius vector with the fixed line PA, WHence 


POLAR CURVES. 179 


this angle is obtuse on the side of the radius vector toward 
the origin, while the subtangent, being also negative, lies on 
the side ofposzze to the origin. 


THE LOGARITHMIC SPIRAL. 


(89) The equation of the logarithmic spiral is 
v=Log.r | 
in which vrepresents the measuring arc and, the radius vector. 
The equation 
may also be put 
into the form F 
a=r 
the relation be- 
tween v and 7 be- 
ing such that 
while wv increases 
in arithmetical 
progression 7 will 
increase In g¢o- 
metrical progres- 
sion. Hence the 
curve may becon- 
structed by lay- 
ing off on the Fig. 36. 
measuring arc the equal distances AJ, dc, ca, de, and so on 
(Fig. 36), and through the points of division drawing the 
radii Pd, Pc, Pa, Pe, and so on, producing them if necessary. 
On these radii lay off the distances PB, PC, PD, PE, and so 
on in geometrical progression, so that 
Ad) yo Ae os EA 
Pie PRPC. Poe eas 
and through the points thus found draw the curve. That 
part of the curve within the circle will be found by laying 
off on the radi Py, Pf, Po, and so on, distances from P by 
the same rule, and thus points of the curve may be found. 


180 DIFFERENTIAL CALCULUS. 


If we make the radius of the measuring circle equal to-1, 
and reckon the arc v from the line PA, then the curve will 
pass through the point A, for when v=o we have 

log. r=o=log. 1 
and if we call a the ratio between PA and PB, we shall 
' have 
PB=a, PG=2*, PD—«@>, PR 2. etc, 
when the exponent 1s always equal to the number of divis- 
ions of the measuring arc, and is therefore represented by 
the arc itself corresponding to the radius vector, whence 
a”* =r or v=Log. 7 to the base a. 
If we differentiate the equation of this curve we have 
adr 


av = | | (2) 
whence (Art. 76) 
rdv  rMar 
tang. PFT= 51 RVI = 


that is 
The tangent of the angle made by the tangent line with the 
radius vector ts constant and equal to the modulus of the system 
of logarithms to which tite system belongs. Jf the system is 
the Naperian, M=1 and the angle PDT is equal to 45°. 
The formula for the subtangent of a polar curve (Art. 78) 
1s 
r* do 
Rar 
and substituting in this the value of eS from equation (2) we 
have (R being 1) 


9 / 


[ 
—==7 MM. 


4 
Suptan— 


If M=r1 then subtang. =>. 
For the value of the tangent we have (Art. 79) 
tang. = 72 +72M?=7V 14+M? 


If M=r1 then ‘a 
tang=rv/ 2 


POLAR CURVES. I8t 


we 


For the subnormal we have (Art. 80) 
arte te 


subnormal ory aT 


If M=1 then 
subnormal =r 
For the value of the normal (Art. 81) we have 


normal=A/7? + “a =rN/1 +373 
If M=1 then 


normal=71/ 2 


These values show that these lines are all in direct pro- 
portion to the radius vector. The same result flows from 
the constancy of the angle made by the radius vector with 
the tangent line. For all the triangles formed by the radius 
vector, the tangent, and the subtangent will be similar to 
each other, at whatever point of the curve the tangent may 
be drawn. The same may be said of the triangles found by 
radius vector, normal and subnormal. Hence these lines 
will always be in proportion to the radius vector. 

To construct a logarithmic spiral for a gzven base, des- 
cribe a circle with a radius equal to a unit of the radius 
veotor, PA, and lay off the arc Ad equal to a unit of the 
measuring arc. Draw the radius vector PB equal to the 
given base; A and B will be points of the curve. Other 
points may be found as already described. 

That part of the curve below the line PA corresponds to 


the negative value of v, and for that we have 
mit 
—@ 
in which when 7=a, v will be infinite. Hence the curve is 
unlimited in both directions. 


SH CPLO NX 


ASYMPTOTES. 


o 


(88) An asymptote to a curve is a line, which the curve 
continually approaches, but never meets. Such a line is 
said to be tangent to the curve at an infinite distance, by 
which we are to understand that the point of contact to 
which the lines approach is beyond any finite limit. 

That this may be the case it is necessary that, at least, 
one of the coordinates of the curve may have an unlimited 
value. Hence when we are seeking an asymptote to a 
curve, our first inquiry must be, whether the equation of the 
curve will admit of such values for the coordinates or either 
of them. If not, there can be no asymptotes. If it willdo 
so for either coordinate, we must substitute that value in the 
equation and ascertain the resulting value for the other 
coordinate. If this resulting value is finite, there is an 
asymptote parallel to the axis of the infinite coordinate; if 
zero then the axis of the infinite coordinate is itself the 
asymptote. But if it should be infinite, then we must resort 
to the following method. 

Find from the equation the values of the coordinates at 
the points where the tangent line intersects the axis, that is, 


182 


ASYMPTOTES. 183 


the distances from the origin. 
These points may be found as 
follows : 

Let A (Fig. 37) be the origin of 
coordinates for the curve SO, and 
let PB be tangent to the curve at 
the point P, of which the coordi- 
nates are x’ and y’. The equa- 
tion of this tangent line is 


@ / 
y—y' =a («—2') 


ax 
If we make y=o we have 
, fire 
x=xX —y i =AB 
If we make x=o we have 
dy’ 


Pon yi Gg oty =AD 


If EC be an asymptote, and the values of x’ and y’ are 
made such as to remove the point of tangency to an infinite 
distance, then AB and AD will become AC and AE. 

If in such case we have finite values for these distances, 
then there will be one or more asymptotes; if there is but 
one finite value, there will be one asymptote parallel to the 
axis of the infinite coordinate. If one be zero then the axis 
of the infinite coordinate is itself the asymptote. If both 
be zero then the asymptote passes through the origin; but 
if both be infinite there is no asymptote. 


EXAMPLES, 


£x.1. The equation of the hyperbola referred to its cen- 
ter and asymptotes is 
xy=M 


in which if « is made infinite y becomes zero; and if y is 


184 DIFFERENTIAL CALCULUS. 


made infinite « becomes zero; hence both axes are asymp- 
totes 


Ex 2. If we consider the hyperbola as referred to its 

center and axes, its equation is 

A2y? =B2 x? — AB? 
where either x or y may be made infinite, and such value 
makes the other infinite also. Hence we take the formulas 
for the points of intersection of the tangent with the axes, 
which give 

A2y! Ay A®y’? —B2x’2 Ae 


cme 


eee pee PRN Me 
X—X ipo ae Bex x! 


and 
ees tea Sawai peta a oe 
AYy AYy of 
both of which values becomes zero, when x’ and y’ are made 
infinite. Hence the asymptotes pass through the origin. 


Ex.3. The equation of the parabola 
y* =2px 
shows that « and y both become infinite together, and hence 
we take 


’ ax’ ! ie , 
and 
las? ety ade GaP ey A 
Va ax roar eis — 5 


both of which values become infinite when x’ and y’ are 
infinite, and hence there is no asymptote to the parabola. 


Ex. 4. If we take the ellipse whose equation is 
A®y? + B22 =A2B? 
we see that neither x nor y can ever be infinite; in fact y 
can never exceed B nor x exceed A; hence there is no 
asymptote to the ellipse. 


£x.5. The equation of the logarithmic curve is 
x=log. y 


ASYMPTOTES. ‘ 185 


It may be constructed by laying off on the axis of abscis- 
sas (Fig. 38) the distances AB, AC, AD, etc., in arithmeti- 
cal progression, and, on the 
corresponding ordinates, the 
distances Aa, Bd, Cz, Dd, etc., 
in geometrical progression, 
and drawing a curve through 
the points thus found. We 
see from the equation that if 
either x or y is infinite on the PIB S Cae Fe 
positive side, the other will be Fig. 38. 
infinite also. If we apply the formula for the intersection of 
the tangent line with the axis we have (Art. 38) 


SNES ners ae 
Vice, paler yo ap (+z) (1) 
and 
be Ue ae ae 
x=x —y eae —y are —M (2) 


We see from these values, that when ’ is infinite x will 
be infinite positively, and y negatively. Hence there is no 
asymptote on the positive side of x. But if « be made 
infinite negatively, y’ will become zero; for the logarithm of 
o is negative infinity, which shows that the axis of abscissas 
is an asymptote on the negative side. ‘The value of y how- 
ever in equation (1) becomes —o%, which is indefinite. 

We learn from equation (2) that the tangent always inter- 
sects the axis of abscissas at a distance equal to M on the 
negative side of the ordinate of the point of tangency. 
Hence the subtangent is constant and equal to the modulus 
of the system to which the curve belongs. If x«’=M, then 
x and y both become zero, and the tangent passes through 
the origin. 

If we put the equation into the form 

ya" 


| DIFFERENTIAL CALCULUS. 
and make x negative it becomes 
. ee 
Daa 


which makes y=o when += 0 


; whence we infer that the 
axis of abscissas is an asymptote to the curve on the nega- 


tive side, as already shown. 


SECTION XI, 


SIGNIFICATION OF THE SECOND DIFFERENTIAL 
COLTFICIEN TF. 


SIGN OF THE SECOND DIFFERENTIAL COEFFICIENT. 


(89) We have seen (Art. 36) that the first differential of 
the ordinate indicates by its s¢gz whether the curve is leav- 
ing or approaching the axis of abscissas; and by its value it 
determines the vate of such approach or departure; that is, 
the tangent of the angle made by the tangent line with the 
axis of abscissas. 

As the point of tangency moves along the curve, the rate 
of its approach to, or departure from, the axis of abscissas 
is constantly changing, and upon the rate of this change will 
depend the direction and amount of curvature of the curve. 

Wherever the curve is situated with reference to the axis 
of abscissas, if its rate of departure is an increasing rate, or 
its rate of approach is a decreasing rate, then the curve is 
convex toward the axis of abscissas; while if its rate of 
departure is decreasing, or its rate of approach is increasing, 
it will be concave toward that axis. 

(90) Now the second differential of the ordinate will 
determine by its sign whether the first is an increasing or 
decreasing function. If the latter is positive and increas- 
ing, or negative and decreasing, its rate of change (that is 

187 


188 DIFFERENTIAL CALCULUS. 


the second differential of the ordinate) will be positive (Art. 
3); but if it is positive and decreasing, or negative and 
increasing, its rate of change is negative. 

Note.— It will be remembered that the sign of the differential and that of its coeffi- 
cient are always the same, since the differential of the independent variable is always 
-uniform and positive. 

(91) If, therefore, the second differential coefficient should 
be positive, the first must be either an increasing positive or 
a decreasing negative function (Art. 3). If the curve is on 
the positive side of the axis of abscissas, it is convex to that 
axis ; if on the negative side it 1s concave. 

(92) If the second differential coefficient is negative, the 
first must be either an increasing negative function, or a de- 
creasing positive one. Hence the curve, if on the positive 
side of the axis of abscissas will be concave, and on the 
negative side convex to that axis. 

(93) To illustrate these principles let us suppose the 
second differential coefficient to be positive, then the first 
must be a positive increasing 
ora negative decreasing func- 99 0 Al 
tion. The curves in Fig. 4o , MIL NG eA 
and 41 answer to these con- p B 
ditions, for from C to D the 
first differential coefficient is Cele ‘ Fy 5 
negative (Art. 36) and de- 40 0 
creasing, while from D to E 
it is positive and increasing in both cases. 

If the second differential coefficient is negative, then the 
first must be positive and decreasing, or negative and in- 
creasing, and we find the curves in Fig. 39 and 42 to answer 
these conditions; for from C to D the first differential coeff- 
cient (Art. 36) is positive and decreasing, while from D to 
E it is negative and increasing in both cases. 

By inspecting these figures we see that for 39 and 4o the 


SECOND DIFFERENTIAL COEFFICIENT. 189 


second differential coefficient has in each case a sign con- 
trary to that of the ordinate, and that both curves are con- 
cave to the axis AB; while in curves 41 and 42 the sign is 
the same as that of the ordinate, and the curves convex to 
Loera nts se Lience 

When the signs of the second differential coefficient and of the 
ordinate are contrary io each other, the curve will be concave 
toward the axis of abscissas; when these signs are althe the curve 
will be convex toward that axts. 

It will be noticed that in all these cases the first differen- 
tial coefficient changes its sign at D where it becomes zero, 
but this does affect the sign nor the value of the second 
differential, for the first may be changing as rapidly, and in 
either direction at the zero point as at any other. 

(94) To illustrate these rules let us take the general 
equation of the circle 


(xa)? +(y—0)?=R? 
in which a is the abscissa and @ the ordinate of the center. 
Differentiating we have 


dy  x—a 
ax y—b 
and 
ay oR 
ee Bs (y—5)8 Fig. 43. 


From which we learn that so long as y is greater than 3 
the second differential coefficient will be negative, while it 
is positive where y is less than 4, or where it is negative. 
Wesee also from the figure (Fig. 43) that above the line DE 
where y is greater than 4 the curve is concave toward the 
axis of abscissas, while between DE and the axis of abscis- 
sas, where y is positive and less than 4, the curve is convex 
toward that axis. Below the axis of abscissas where y is 
negative the second differential is still positive, while the 


190 DIFFERENTIAL CALCULUS. 


curve is concave toward the axis. -All of which corresponds 
with the rule. : 

In the case of the parabola referred to its vertex and 
axis we have 


oy a pe 
hoe eee 
a fraction whose sign is always contrary to that of y,; hence 
the curve is always concave towards the axis of abscissas. 
The same may be said of the ellipse referred to its center 
and axes from whose equation we have 


Tea) MBs 
adx® ~ A®y3 


In the case of the hyperbola referred to its center and 
asymptotes we have 
eye Iey, 
; aa 


a fraction whose sign is always the same as that of y. 
Hence the curve is everywhere convex toward the axis. 


VALUE OF THE SECOND DIFFERENTIAL. COEFFICIENT. 


(95) The curvature of a curve at any point is the én- 
dency of the tangent line at that point to change its direc- 
tion, as the point of tangency is moving along the curve, in 
obedience to the daw of change derived from the conditions 
which govern the movement of the generating point. 


Note.— The curvature then ofa curve is zof ‘‘ its deviation from the tangent,’** nor 
‘‘its departure from the tangent drawn to the curve at that point,’’t nor is it ‘‘the 
angular space between the curve and its tangent,’’} nor isit any acfwadZ change in the 
direction of the tangent line as the point of tangency moves along the curve ; nor does 
it depend on any such change, but upon the LAw which governs the movement of the 
generating point ; for itis this law which fixes the ¢ezdency of the tangent to change 
its direction and this tendency is the curvature. Hence in estimating the curvature of 
a curve at any point, we consider that point a/ove and seek, zof any actual movement 
of the generating point, but the Zaw which controls Zt, 


*Loomis. tDavies. {Church. 


SECOND DIFFERENTIAL COEFFICIENT. I9t 


Hence if several curves as CD, C’D’, C"D" (Fig. 44) have 
coincident tangeuts AB at the point A, and if we suppose 
the point of tangency to be at any instant moving along the 
curve, Carrying with it its own tangent _ A B 
line, that one whose tangent line at — > 
the moment of coincidence is chang- Ly. N 
ing its direction most rapidly will Co Cie (ee Be 7 
have the greatest curvature at that 
point. For the raé of change in the «Fig. 44. 
direction of the tangent is the measure of its zendency to 
change. 

Since the first differential coefficient indicates the drec- 
tion of the tangent to a curve, by means of the tangent of 
the angle made by it with the axis of abscissas; the second, 
which is simply the rate of change in the first, will indicate 
the rate at which the tangent of that angle is changing its 
value.. Now as between two curves at common tangent 
point, that curve in which the tangent line tends to change 
its direction most rapidly, will be the one in which the 
tangent of the angle made by that line with the axis of 
abscissas will also tend to change z¢s va/ue most rapidly, and 
will, therefore, have the greatest curvature, while if these 
tendencies are equal the curvatures are equal, and this 
will be indicated by the equality of the second differential 
coefficients. 


SECT LONE 


CURVATURE OF LINES. 
THEOREM. 


(96) Zhe curvatures of different circles are inversely propor- 
tional to their radit. 

The curvature of a circle is the same at all points of the 
circumference, and all circles having the same radii have 
the same curvature. 

Since the change in the direction of the tangent, as the 
point of tangency moves around the curve is constant, its 
actual change of direction for any given movement of the 
point of tangency, will always be in proportion to its “z- 
dency to change, multiplied by the length of the are over 
which the movement 1s made, and may, therefore, be repre- 
sented by that product; and hence the zendency to change or 
curvature will be equal to the actual change divided by the 
length of the arc. 

Now the change in the direction of the tangent is equal 
to the angle contained between its two positions, which is 
the same as that contained between the two radii drawn to 
the extremities of the arc. Calling this angle v and the 
length of the arc a, we shall have 


v 
curvature aes 


192 


CURVATURE OF LINES, 193 


If now we have two circles, which we will call 0 and o’, 
whose radii are 7 and 7’, and the angles at the center for the 
same length of arc a are v and 7, we shall have 


7) 


< 
curvature of (Sects 


t 


o 


curvature of 0’ Bye 


hence 
curvature of 0: curvature of 0’ ::v: v7 (1) 

but ; 

2000s Ger 
and 

Time lathe Fer ey vat 
whence 

y.2nr=v . arr’ 
or 


Substituting this ratio in proportion (1) we have 
curvature of 0: curvature of 0 ::7':7 


Ore ew Le 


CONTAGE OF CURVES, 


(97) When two curves have a common point, the coordi- 
nates of that point must satisfy both their equations. This 
will generally be a point of zztersection, and not a point of 
contact; and is all that can be secured by having but one_ 
condition common to the two curves. 

If they are at the same time tangent to each other, at the 
common point, then another common condition is imposed 
and there is a contact of the first order. 

The condition required in this case is, that, for the point 
of contact, the first differential coefficients shall be the same 
for the equations of both curves. For since the curves are 
tangent to each other, they have a common tangent line, and 


194 DIFFERENTIAL CALCULUS. 


the first differential coefficient, which determines the angle 
made by this line with the axis of abscissas, must be the 
same for both equations. 

If, besides this, the curves are required to have the same 
curvature at the point of contact, this will introduce a third 
condition, which is, that the second differential coefficients 
shall be the same for both equations (Art. 95). 

For the second differential is the rate of change in the 
first, which gives the direction of the tangent line, and the 
rate of change in this direction is the curvature. This isa 
contact of the second order. ; 

If now it is required, in addition, that the rate of change 
in the curvature should be the same in both curves at the 
point of contact; we must introduce a fourth condition, 
viz., that the ¢izrd differential coefficient should be the same 
in both equations. ‘This would be a contact of the third 
order. And thus the order of contact would become higher 
for every new condition introduced common to both curves, 
and every new agreement between the successive differential 
coefficients. 

If then we wish to find the order of contact of two given 
curves, we first combine their equations, and determine their 
common point if they have one. For this point the varia- 
bles will have the same value in both equations. If the 
values thus found being substituted in the first differential 
coefficient of each equation, reduce them to the same value, 
there is a contact of the first order; that is, they have a 
. common tangent line at the common point. 

If they also reduce#the second differential coefficients of 
the two equations to the same value they have a contact of 
the second order, and so on for the successive differential 
coefficients ; the order of contact being determined by the 
number of coefficients that successively become equal by 
the substitution of the values of the common coordinates. 


CURVATURE OF LINES. 195 


EXAMPLE, 


(98) To illustrate this rule let us take the two equations 
4y=x?—4 (1) 
and 
Tiler SR Pret (2) 
from which we obtain by combination 
y=—I and «=o | 
indicating that both the curves pass through the point of 
which these are the coordinates. We have also by differen- 
tiating twice — for equation (1) 
aes ary 


dz 2 M0 age t 
and for equation (2) 
‘ay x a*y I xe 
da ya ey 


Substituting in these differential coefficients the values of 
x and y just found, we have the first differential coefficients 


ay es ay 6 
> = _=o and 5G-=— =o 
Cane ax Neen! 
and the second differential coefficients 
a*y a*y I a 
4} and —5 = FT 
i let le ae Mam (YT eee 


from which we infer that at the point whose coordinates are 
x=o and y=—1, the curves have a contact of the second 
order. We also see from the value of the first differential 
coefficient that at that point the tangent to both curves Is 
parallel to the axis of abscissas. A little investigation 
would show that the first curve is a parabola, and the sec- 
ond a circle tangent to the first at its vertex. 

(99) The constants which enter into the equation of a 
curve determine the conditions.which govern the movement 
of the generating point for that kind of curve; which must 
fulfil as many conditions as it has constants. . Thus the cir- 


196 DIFFERENTIAL CALCULUS. 


cle whose general equation contains three constants, must 
fulfil three conditions, namely, two in the coordinates of the 
center, and one in the length of the radius. The ellipse 
must fulfil four conditions, namely, the coordinates of the 
center and the lengths of the two axes. x. 

(100) Now if one curve be given complete by its equation 
with fixed values for its constants, and another with con- 
stants which are indeterminate, and capable of being adjusted 
to any given conditions, we may easily assign such values to 
them as will cause the curve to fulfil such conditions as 
may be required of it. We may, for instance, require the 
curve to pass through a given point in a given curve. This 
will require that the same variable coordinates shall satisfy 
the equations of both curves for that point. We may also 
require them to have a common tangent at that point; this 
will require the constants to be so adjusted that the first 
differential coefficients of the two equations shall be equal. 
If there are three or more constants in each equation we may 
require such values as will cause the second differential 
coefficients to become equal also, thus producing an equality 
of curvature, or a contact of the second order, at the com- 
mon point. And thus we may continue until the order of 
contact is one less than the number of constants to be dis- 
posed of. 

(101) In order to make this adaptation of the second curve 
to the first we must consider its constants, or as many of 
them as will be required for the purpose as unknown quan- 
tities (Art. 4) and construct as many equations as may be 
required to determine them. 

These equations are derived from the conditions to be 
fulfilled by the constants. Thus the first which requires 
that the second curve shall pass through a point of the first 
will generally be met by the proper adjustment of a single 
constant; and an equation formed by substituting in that of 


CURVATURE OF LINES, 197 


the curve to be adjusted the values of the coordinates of the 
designated point, and also the values of the known con- 
stants, will determine the value of the unknown constant. 

If itis required that the two curves be tangent to each 
other, we must adapt the values of “wo constants to this 
condition, and this is done by substituting the same values 
of the common coordinates, and of the remaining constants 
in the fs¢ differential coefficients of the two equations, and 
placing them equal to each other, thus forming a second 
equation. A contact of the second order may be secured 
by fixing the value of a third constant in a similar way by 
means of the second differential coefficients of the two equa- 
tions. 

The values of these constants thus determined being substituted 
in the general equation of the required curve, will produce an 
equation of one that will fulfill the required conditions ; that is, 
one that will intersect at a given point, or have a contact of 
a required order. 


EXAMPLE. 


(102) To illustrate these principles let us take the equa- 
tion of the ellipse referred to its center and axis 
A2y?+B2x? =A*B? 
and the general equation of the circle 
(a a)* y= )ee=R2 (1) 
in which the constants are arbitrary and may be adapted to 
any prescribed conditions. Suppose we say that the cir- 
cumference shall pass through the upper extremity of the 
conjugate axis where 
x=o and y=B 
This being but one condition will require the adaptation 
of but one constant. Let this be a, while we make R=A 
and d=o. 


198 DIFFERENTIAL CALCULUS. 


Then substituting these values in equation (1) we have 
(o—a)* +(B—o)? =A? 


or 
a*+B?=A?*® 


a=LV A? —B? 
and the equation of the circle becomes 
(2 V A? —B2)? +y2 =A? 
the center being in one of the foci—the plus value of the 
radical corresponding with the focus on the positive side of 
the center. 

If we add another condition, namely, that the curves shall 
be tangent: to each other at the same point, we must adapt 
the value of fwo-constants to these two conditions. Let 
these constants be @ and 4, and make R=2B. Then we 
must construct an equation between the first differential 
coefficients of the curves; that is 

B*’x x—a 
Apes oy (2) 
Substituting the values of x and y as before we have 


whence 


B*o o-—a 


A?B B—é 
hence 
a=o 
and substituting these values in equation (1), we have 
(B—)? =4B? 
whence 
b=—B 


and the equation of the required circle becomes 
Hes tN Pt 
the center being at the lower extremity of the conjugate 
axis where a=o and J=—B. 
If now we add a still further condition there shall be a 
contact of the second order at the same point we must adapt 


CURVATURE OF LINES. 199 


the values of ##ree constants to that condition, by forming a 
third equation, between the second differential coefficients, 
thus ¥ 


ay” (%:=a)* 
B4 Tax? TG? 
A? y3—_ y—d Sy, (3) 


Substituting, as before, the values of x=0 and y=B in 
equations (1), (2), (3), we have three equations from which 
to determine the values of the three constants; thus 

(o—a)? +(B—2)? =R? 
et aan. c 
A’B B—d 
(o—a)* 
Be 11 (B32 
AZ Beebo 
From the second we obtain 


a=o 
From the third we have 
DAste a 
a a ae 
and substituting these values in the first we obtain 
A2 
race 


and the equation of the circle becomes 
[eee ey any! 
et ins 
the radius being equal to half the parameter of the conju- 
gate axis of the ellipse, and the center being in that axis 
prolonged in a negative direction. 

(103) In this last case we have the highest order of con- 
tact of which the circle is capable, and hence the circle is 
called the osculatrix to the ellipse; or is said to be oscula- 
tory to it. 

An osculatrix toa curve is one which has the highest order of 


200 DIFFERENTIAL CALCULUS. 


contact wrth tt, that any curve of the same kind as the osculatrix 
can have. ; 

Since the number of constants limits the number of con- 
ditions that can be assigned to a curve, and since the pass- 
ing of the curves through the same point is one condition, 
the order of contact can only be equal to the remaining 
number of possible conditions ; namely, the number of con- 
stants, less one, which enter into the general equation ; and 
this will be the same as the order of its highest differential. 


EXAMPLES. 


(104) x. 1. To find the equation of the circle oscula- 
tory to the parabola, whose equation is 
y" = 4o0 (1) 
at the point where the coordinates are 
x=1 and y=2. 
Differentiating this equation we have 
dy :- 2 a*y 4 
wa hes 38 d _— 


dx y *P° “axe y 
whence 
2s ei 
ye ye 
and 
x—ay* 
ieee hal pe 4 
or 
I—a 
pet rae (2) 
and 
I—a\* 
fee =) (3) 


CURVATURE OF LINES. 201 


Also from the general equation of the circle we have 
(1—a)? +(2—0)? =R? (1) 
and from these we find 
R*=32, a=5, b=—2 
and the equation of the circle osculatory to the ERE at 
the given point is 
(x—5)? +(y+2)?=32 
Fx, 2. ‘To find the circle osculatory to an equilateral 
hyperbola whose equation is 
xy=8 
at a point whose coordinates are 
y=4 and *x=2. 
By differentiating we have 


dy J Net 
ax aie 
and 
DE Ye 27, 
ax® x2 : 
and from the general equation of the circle we have 
(2 SecA) ae (1) 
=a 
“ig ee (2) 
2—a\* 
iii 
tga bots (3) 
4—b 


from which we obtain 


a 


4 2 
and the equation of the required circle will be 


: 13 We es 
Capes) = 
Fx, 3. Find the equation of the circle osculatory to the 


curve whose equation is 


202 DIFFERENTIAL CALCULUS. 


4y= x? —4 
at a point whose coordinates are 
x=0 y=—-I 


RADIUS OF CURVATURE. 


(105) Since the curvature of a curve at any point is the 
same as that of its osculatory circle at that point, we call 
the radius of the osculatory circle the radius of curvature of 
the curve. And since the formulas for the equation of the 
osculatory circle may be applied to any point of a given 
curve, we may consider them as expressing the general con- 
ditions required of the osculatory circle. 

These formulas, as we have seen, are 


(x—a)’ +(y—0)? =R? (1) 
ay  (x—a 
dx y—b (2) 
dy” 
Bose LT aa? | 
axe” y= | (3) 
the two last may be written 
ay 
sare Focarrmee a Us Foot , (2) 
and 
ax* +dy* 
aa ae a ary (3) 


If we represent the coordinates of any given point in a 
curve by x’ and y’, then for the osculatory circle we must 


have 
x=x", yay, ve Coden y 
i eUEN ee ACS VEY 
The quantities @ and 4 represent the coordinates of the 
center of the osculatory circle, and R is its radius. 
If we substitute in equation (2) the value of y—d, we 


have 


CURVATURE OF LINES. 203 


ae au e)) 
ax a*y 
whence equation (1) becomes 


x—a= 


dy* “2 ee Ce tory" =R 
ax* a ay iz wes ar 
from which we have 
3 
ie (dx? +dy?)? 
ee dx *y (5) 


which ts the general expression for the value of the radius of 
curvature in terms of quantities belonging lo a given curve. 
If we denote the length of the curve by w we shall have 
aus 


decd y 


(106) Since the curve and its osculatory circle have a 
common tangent, they will also have a common normal; 
and as-the normal to the circle passes through the center, 
the normal to any curve at any point will pass through the 
center of the circle osculatory to it at that point. 

This is also shown from equation (2) which is 

Le Ee 

ST 
x and y being coordinates both to the given curve and to 
the osculatory circle at the point of contact, and a and # the 
ceordinates of the center of the circle. 


: ay. 
For since ope Us the tangent of the angle made by the 


tangent line with the axis of abscissas, we shall have 
a rool 
See 
for the tangent of the angle made by the normal line with 
the same axis. But when astraight line passes through two 
points — «x and y being the coordinates of one, and a and 


the coordinates of the other—the tangent of the angle 


204 DIFFERENTIAL CALCULU». 


made by that line with the axis of abscissas will be 
=o 
x—a 
through the first point will also pass through the second — 
that is, the center of the osculatory circle. 
And since from equation (3) we have 
8 


ax* 


and hence the normal to the curve, since it passes 


Poe pn 
(y—0)2 =~ (14 


the value of the first member of the equation will be essentially 


Dy 
~~ 


negative, and hence we infer that y—4 and on must have 
ove 
ye 8 hegative, 6 will be less 
than y, and, if positive, it will be greater. In the first case 
the curve will be concave toward the axis of abscissas, and 
6 will be between the curve and that axis; while in the 
other case the curve will be convex toward the axis of 
abscissas, and 4 will be deyond it. Hence the center of the 
osculatory circle will be on the concave side of the curve. 
(107) To find the general expression for the radius of 
curvature of the parabola, we differentiate its equation twice 
and obtain 


contrary signs. So that if 


ydy=padx 
and 
yd*y tay? =o 
whence 
pax 
ay =—— 
ATs 
and 
acy janie 
iy ee 
cf y yp 
Substituting these values in the formula we have 
3 
2Ixey z 
VR od i ESS 3. 3 
eS) pate tel _coetpn! 
— prdx? > — p*dx* Pe —p* 


—dUx 
ys 


CURVATURE OF LINES. 205 


or, the cube of the normal (Art. 56) divided by the square 
of half the parameter. 
If we make x=o we have 
R=p 
or half the parameter for the radius of curvature at the 
vertex. If we make x=} we have 
R=pvV/ 8 
for the radius of curvature at the point where the ordinate 
through the focus meets the curve. As every other value 
of R is greater than that where xo it follows that the 
greatest curvature of the parabola is at the vertex. 


(108) From the equation of the circle we have 


Ba 
ady=— 
: y 
and 
R&dx* 
a? y=——_ >? 
y 
and substituting these values in the formula we have 
3. 
(2 span} ‘ 
CO ee 213 
sel Ve Ot? ext | 
peg Re rai aaa 


ye 
the radius of the circle as it should be. 
(109) From the equation of the ellipse we have 
B*xdx 
TS Or 


and 
Atay? +Bedx? 
De es TERE 
ha 
or substituting in the last equation the value of zy” we have 
Baa 
A®ys “ 
These values being substituted in the formula 


Wi 


206 DIFFERENTIAL CALCULUS. 


3 


Beatin (A eae 


‘3 ye A4 
Bie me aa he bee 
A2y3 ake 


which is equal to the cube of the normal divided by the 
square of half the parameter as in the parabola. 
If we make x=A we have y=o and 


B2 
ae 

If y=B then x=o, and we have 
A2 
US tre iy: 


Hence the radius of curvature of the ellipse at the princi- 
pal vertex is half the parameter of the transverse axis — 
that is the ordinate through the focus. At the vertex of the 
conjugate axis, the radius is half the parameter of that axis 
(Art. 102). 

(110) The equation of the hyperbola referred to its center 
and asymptotes gives 


) ae aye 
oN Sy eager 
and 
r 2axay 
Sa oo 


Substituting these values in the formula we have after 
reducing. 


. 


aol 
ages 


_(atty?)#_ala?+ty?) 
205) AL 

In the equilateral hyperbola, this value becomes equal to 
the cube of the radius vector divided by the square of the 


semi-axis. 


R 


SECT LON XII SE. 


PICOL EF L465 


(fff) If we suppose a circle to roll along the concave 
side of a curve, being always tangent to it, and at the same 
time varying the length of its radius so as to be osculatory 
also, its center will describe a curve which is called the 
evolute of the given curve; and its variables will be the 
coordinates of that variable center. In other words, the 
evolute of any curve is the /ocuws of the centers of all the 
circles that can be drawn osculatory to that curve. 

The relation between the variables of the evolute can be 
determined and its equation found from the equation of the 
given curve, and the first and second differential coefficients 
derived from that equation; since these determine the posi- 
tion and length of the radius of curvature, and consequently 
the place of the center of the osculatory circle. 

Since the coordinates of the point of tangency and the 
first and second differential coefficients are the same for the 
given curve and for the osculatory circle, we can at once 
determine two of the properties of the evolute. . 

(112) The first of these properties is, the radius of the 
osculatory circle is tangent to the 
evolute. 

Let AC (Fig. 45) be any curve, 
and let ¢c be the center of the oscula- 
tory circle for the point A, while’, 
c’,c’’ are the centers of the oscula- 
tory circles corresponding to the 


207 


208 DIFFERENTIAL CALCULUS. 


points A’, A”, A”. Then the curve cc” passing through these 
centers. will be the evolute, and any radius as A’c’ will be tan- 
gent to it at the point ¢’, the center of the osculatory circle. 

The equations of conditions (Art. 105) may be put into 
the following form 


(x—a)? +(y—4)? =R? (1) 
(x—a)dx+(y—b)dy=o (2) 
(y—6)d®y +dy® +ax® =o (3) 


and in this case a, 4, R, x, y are variables; x being indepen- 
dent and dx a constant quantity; while x and y are coor- 
dinates of the given curve, and of the osculatory circle at 
the point of contact, and @ and 6 coordinates of the varia- 
ble center of the osculatory circle, that is, of the evolute, 
and are functions of x and y. 

From these equations, as we have seen (Art. 105), R may 
be determined for any point in the given curve by eliminating 
a and 4 considered as constants. But for the evolute curve 
we must consider them as variable coordinates; and hence 
under that supposition if we differentiate equations (1) and 
(2) we have 

(«—a)dx+( y—b)dy—(x—a)da—(y—b)db=RAR_ (4) 


and 
ax? +dy? +( y—b)d* y—da.dx—db . dy=o (5) 
Subtracting equation (2) from (4) we have 
—(x—a)da—( y—b)db=RAR (6) 
and subtracting (3) from (5) we have 
, —da.dax—db .dy=o (7) 
whence 
db = ax 
da ay (8) 


joey, 
but peiaye® the tangent of the angle made by the normal 


line to the curve, at the point whose coordinates are x and y, 


; db 
with the axis of abscissas: and “om the tangent of the an- 


EVOLUTES. 209 


gle made by the tangent line to the curve at the point whose 
coordinates are a@ and 4 with the same axis. But x and y 
are coordinates of the given curve, and a and 2& are coordi- 
nates of the evolute, and, of course, of the center of the 
osculatory corresponding to the point (*.y) on the curve, 
and through this center the normal line must pass (Art. 
106); and since both the normal to the curve (or radius of 
curvature) and the tangent to the evolute pass through the 
same point, and make the same angle with the axis of 
abscissas, they must be one and the same line; and hence 
the proposition. 

(113) The other property referred to in Art. 111 is 

The difference between the length of the evolute curve and the 
radius of curvature, measured from the same potnt ts etther zero 
or a constant quantity. 

From equations (2) and (8), of the preceding article, we 
have 


ada 
x—a=— (y—8) (9) 
and substituting this value of x—a in equation (1) we have 
aa” da” +dab* 
BIST Cape tae 9 bed oa eet) hermegere vem LS) 
From equation (9) and (6) we have 
aa” da? +d? 
en at) U) 0 )av— Ra Rae 7-0) 
which being squared gives 
1a" a0" )* 
(yo EY RoR 


and this being divided by equation (10) gives 
da® + ab? =a R* 
If we designate the length of the evolute by w we shall have 
du? =da* +db* 
whence 
adu*® =a R® 


210 DIFFERENTIAL CALCULUS. 


or 
du=dR or €R—du=o=a(R—xz) 
hence R—vwz is a constant quantity and 
R=u-+c 
If wz=o0 we have 
R=c 
and hence ¢ is equal to the radius of curvature at the begin- 
ning of the curve, and R is at all times equal to the length 
of the evolute to the point where. R is tangent plus the con- 
stant ¢. 

If, therefore, we suppose a cord to be fastened at B (Fig. 
45) and drawn tight around the curve AB and then unwound 
from A, the end of the cord will describe the curve AC of 
which the curve AB is the evolute. For the cord will be at 
all times tangent to the curve from which it is unwound, and 
also the momentary radius of the curve AC for the point at 
its own extremity, and consequently normal to the curve at 
that point; while the length of the cord from the point of 
tangency to its extremity in the curve AC is equal to the 
distance from the same point to the origin at A measured 
along the curve AB. 

({14) To find the equation of the evolute, we must com- 
bine the equation of the osculatory circle with that of the 
involute in such a manner that x.y and R shall disappear 
and leave an equation containing only a and @ as variables. 

This will require four equations, and these are obtained 
from the equation of the involute, the general equation of 
the circle, and those formed by placing the first and second 
differential coefficients of each of these equations respec- 
tively equal. 

Thus if we take the equations of condition (Art. 105) 


(2a) eR (1) 
ay ay  x«-—a@ 
7 ay ey A (2) 


EVOLUTES. 21ft 


° 
wo 


a 

ax* +dy* my Ie ax? 

a zy Oe yas ee y—b (3) 
and then differentiating the equation of the involute twice, 
we find the values of the same differential coefficients and 
make them equal to the second members of equations (2) 
(3); then eliminate x, y and R, the resulting equation is that 
of the evolute. 

Since R is contained in only one equation, we omit that, 


eae aad 


as the remaining three are sufficient for eliminating x and y, 
and for the resulting equation. 
(115) To find the equation of the evolute to the parabola. 
The equation of the parabola is 


Vy = 2px (1) 
from which 
dy p 
ae. (2) 
and 
ax premier (3) 


Placing these differential coefficients equal to those of the 
general equation of the circle, we have 


Pe XE 
nie beer (4) 
and 
dy* 
es 
Pom ye (5) 


Dividing equation (5) by equation (4), and substituting for 
y? its value ffom equation (1), and reducing, we have 

a=3x+f (6) 
and substituting the values of @ and y in equation (4) we 
have after reducing 


212 DIFFBRENTIAL CALCULUS. 


and substituting in this the value of x from equation (6), 
and squaring, we have 
3 
po PY Sop) 
3°P iP 
which is the equation of the evolute of the parabola. 

If we make =o we have a=, which is the center of the 
osculatory circle for the vertex. If we transfer the origin to 
that point we have 

a=ptad and d=0 


hence 


Since every value of a’ gives two 
equal values for J with contrary 
signs, the curve of the evolute ACE 
(Fig. 46) is symmetrical about the 
axis of abscissas. If a’ ‘is negative 
then J’ is imaginary, and hence the 
curve commences at C,a point inthe 
axis of abscissas at a distance from “ 
A equal to — that is, at double the Fig. 46. 
distance of the focus, or half the parameter. 


(116) To find the equation of the evolute of the ellipse. 
For this case we have 


A®y? + B242 =A2B? (1) 

(See Bin ea 

cog A®y yb (2) 
ay” 

oo ae nie SEE?) 

ax? A®y3 ~~ y— (3) 


From equations (2) and (3) we have 


dy” 

Qaro ye 

_ A®y(x—a) 2s yr ax? 
pa Bea BA 


EVOLUTES, 213 


whence 
Btx? 
X—a +E) Wid Coa aay, 
x Be ei Be 
whence 
AA +Btx? 
B2(x—a)=2( Ad ) 
whence 
AB? x—AtB*a=Atay®? +Btx3 
whence 
A? x(A®2B?—A®y?)=A4B2a+B4x3 
whence 
A®B? xe =A4B*atBitx?3 
whence 
prey 
Serekag x8 (4) 


Substituting this value of @ in equation (2) we have 


Aeoeo e O) Re 


Kaa en 


A4 A’y 
whence 
A*—(A?—B? ae? (y—6)B* 
iN: y 
whence 
Aty—A2®x*y+B*x*?y=A*B*y—A?B?O 
whence 


OR ato Mele a o 
Substituting for x? its value from equation (1) we have 
A2B2—A2y? 


UG ie eae ea eam d 
whence 
A?—B? F 
b= — pane) (5) 


Making A*—B* =c? we have 
2 


fea c 
a=ix® and ects a a¥ BA 


214 DIFFERENTIAL CALCULUS. 


eetits a 
and making Fai and BR” we have 


or 


1 
3 a\3 y by3 
Kalp)) ond penal) 
Writing the equation of the ellipse under the form 


2 2 
x fe ea 
AE eB te 


mene 2 
and substituting the values of cy and = just found, we 


have 


2 
ay b ask 
(7) +(;) = (6 
which is the equation of the evolute of the ellipse in which 
a and # are the variable coordinates, and m and z the con- 
stants. If we make a=o we have J=-+z, and if 6=o we 


have a=-+m, which shows that the form of the evolute is 
symmetrical with both axes of the ellipse. But 


ak eri se 
Amat eae aes 
and subtracting this from A we have the radius of curvature 
B? 


at the principal vertex equal to A as we have already seen 
(Art. tog). Similarly we find the radius of curvature at the 


nS 
vertex of the conjugate axis to be 3: 


If we differentiate equation (6) twice we have 


hae ih 
aC) +-(<) yr 8 


whence 


EVOLUTES. 215 
=i 
(2) 
eal | bare gol 
ab m\m m2 ( aa 3 


da -4 n\bm 


and 


ay 


( ee 20 


FS ae 


ada n 


ae 


n\n n 


whence 
me —4 
Uta A eh IY De 
AESPUG) eee ex 
da” Aes 
de 

Since the numerator of the second differential coefficient 
is always positive, it will have the 
same sign as the denominator, which 
is the same as that of J, and hence 
the curve is everywhere curvex 
toward the axis of abscissas. The 
first differential coefficient becomes 
zero when ao, and infinite when =o, hence both axes 
are tangent to the curve, asin Fig. 47. 

If we make A=B, then c=o, and also m=o and n=a, 
hence a and 2 in equation (7) will also become zero as they 
should, since in case of the circle the evolute is reduced to 
a point — the center. 


SECTION XL yve 


ENVELOPES. 


(117) Suppose two lines, AB and AC (Fig. 48), be drawn 
at right angles to each other, and a third line ed to move in 
such a manner that its extremities @ and e shall be con- 
stantly in these axes, while its length remains unchanged; 
so that while the extremity e¢ arrives c 
successively at the points ¢,¢", the 4g 
extremity @ will arrive at the corres- 4g 
ponding points a’, a”. di’ 

During this movement those points 
of the line near the extremity @ will 
move in the direction more nearly A ae B 
parallel to the axis AC than the lne Fig. 48. 
itself 1s, and will consequently fall zzthzn its first position, 
while the points near the extremity e will move in a direc- 
tion more nearly parallel to the axis AB than the line is, and 
will consequently fall z¢houtits first position. But between 
these extreme points there is one that ends te move in the 
direction of the line ttself. ‘ 

This point does not, of course, remain fixed on the line, 
but moves from one extremity to the other as the line changes 
its position and direction, always occupying that place in the 
line which at the moment does not tend to move out of it 


216 


ENVELOPES, 217 


towards either side. Zhe curve described by this point ts the 
envelope of the curve. 

Again let AB (Fig. 49) be the transverse axis of an 
ellipse, and CD its conjugate axis; and suppose these axes 
to vary to any extent under the condition that the area of 
the ellipse shall remain constant. 7 

Then as AB decreases CD will Boe 


increase at arate corresponding with 
this condition. When the curve 

) Seam 
thus commences to change its shape, | C | " 


a point near the extremity A will —(4 Fj 
tend to move in a direction more NAL fp LY 
nearly parallel to AB than the tan- re 

gent to the curve at that point is; ip” 

while a point near the extremity C Fig. 40. 

will tend to move in a direction more nearly parallel with 
CD than the corresponding tangent line is. Now between 
A and C there is a point in the curve that tends (as the 
axes are changing) to move exactly in the direction of the 
tangent to the ellipse at that point. 

As the curve changes its shape and position this point 
will also change z¢s place on the ellipse, keeping always 
where its zeudency 1s in the direction of the tangent to the 
ellipse as it is at the moment. The movement of the point 
will be continuous, and it will generate a curve which will 
be the envelope of the ellipse. 

(118) Since the point on the given curve which describes 
the envelope always tends to move in the direction of the 
momentary position of the tangent to the curve at that 
point, and since any generating point always tends to move 
in the direction of the tangent to its own curve, it follows 
that the given curve and its envelope will have a common 
tangent line wherever the generating point may be at the 
moment during the formation of the curve. Thus in the 


218 DIFFERENTIAL CALCULUS. 


last illustration, the ellipse, in every stage of its change, will 
be tangent to the envelope at that point of the curve just 
then generated. 

(119) Av envelope to any line, ts another line generated by that 
point of the given line, which lends to move in the direction of the 
tangent, whenever tts position or shape ts made to change by chang- 
ing the constants of tts equation, or any of them, tnto variables. 

An envelope is not always produced by this change of the 
constants, for it may be that no point of the given line will 
tend to move in the direction of its tangent; as in the case 
of an ellipse where both axes are increased. 

In general, there will be an envelope only where the suc- 
cessive positions of the line corresponding with minute 
changes in the constants, will zzéersect each other; for while 
the generating point of the envelope tends to move in the 
direction of the tangent, the points on each side of it will 
tend to move away from the tangent in opposite directions, 
hence the next position of the changing line will cross the 
previous one near the generating point of the envelope. 

(120) If in any equation of a line the constants are made 
to vary in value, it is evident that while the curve or line 
remains the same in kind, its shape and position may assume 
every possible form and place within the lhmits determined 
hy the law of variation imposed upon the constants of the 
equation. 

If we take for example the ellipse, and consider A and B 
in its equation as independent variables, then 

A®y? +B2x? =A*B2 
will represent an infinite number of ellipses of every possi- 
ble size and proportions subject to but two conditions; 
namely, the axes must both coincide with the axes of coordi- 
nates. If we make A and B dependent on each other we 
limit the system of ellipses by the condition thus introduced, 
but still their number is infinite. If we introduce the still 


ENVELOPES. 219 


further condition that the values of x and y shall be confined 
to those points of the system which tend to move in the 
direction of the tangent, while A and B tend to change their 
values, the first differential coefficient will not be affected by 
such tendency in A and B, and hence will be the same a¢ 
those points whether they are considered as variables or con- 
stants. So then if we take the differential of the equation 
with respect to them only as variables, and make it equal 
to zero, and incorporate it with the original equation, we 
put this limit on the values of « and y, which will then 
only apply to points in the envelope. The equation will, 
therefore, be that of the envelope itself —that is, zzstead of 
representing every point tn one ellipse, tt will represent one point 
an each quadrant of every ellipse thatcan be formed under the 
given conditions. | 

To find the equation then of an envelope we differentiate 
the equation of the given line with reference to such only 
of the constants as are considered variable for the time 
being, and place that differential equal to zero. ‘The values 
of the constants determined from this equation, and the 
conditions of relation among themselves, being substituted 
in the given equation, will produce one that will be inde- 
pendent of the variable constants, and this will be the equa- 
tion of the envelope. 


EXAMPLES, 


(121) For the first example, let us take the general equa- 
tion of the circle in which R and @ are constants, while a is 
considered as a variable. Now since the values of x and y 
are to be confined to those points of the circle which tend 
to move in the direction of the tangent while. a varies, it 
will make no difference whether we differentiate with re- 
spect to x and y only, or with respect to a also. Differenti- 
ating in both these ways we have 


220 DIFFERENTIAL CALCULUS, 


(«—a)dx+(y—b)dy=o 
and ) 
(«—a)dx—(x—a)dat+(y—b)dy=o 
making these differentials equal, and cancelling like terms 
we have 
—(x—a)da=o (1) 
which we should have obtained at once by differentiating 
with respect to a alone, considering all the rest as constants. 
From equation (1) we have 
x=a 
and this value substituted in the general equation gives 
y—b=£R or y=dER 
If we take the positive value for R, this is the equation 
of a line DE (Fig. 50) | 


parallel to the axis of ab- ©} = 
scissas at a distance equal 

to that of the centers of 

the system of circles plus 

the radius, and hence tan- , | p/ E’ 
gent to them all on the Reger 


upper side, and is genera- Fig. 50. 

ted by the highest point of the circle as it moves from D to 
E, as @ varies in value; that point being the one that tends 
to move (and in this case does move) in the direction of the 
tangent to the circle drawn through it. If we take the neg- 
ative value of R, the equation represents the line D’E’ tan- 
gent to the system of circles on the lower side. 


(122) If we take the same equation and consider a and é 
both as variables, we must establish a relation between them 
in order to make them both functions of « and y. Let this 
relation be expressed by the equation 

a®* +6? =? (1) 


then the two equations will represent a system of circles 


ENVELOPES. 


(Fig. 51) whose centers lie in the 
circumference of another circle 
whose radius is equal to ¢, and its 
center is at the origin. 

Differentiating the general equa- | 
tion of the circle with respect to a \ 
and 4 only we have 


Lb 
= (x= a) =(y-8)G=0 


whence Fig. 51. 

GO. Sts E 

da —-y—b 
We may now substitute for 4 its value obtained from equa- 
tion (1); or we may consider it as a function of @ in that 
equation and substitute the value of the differential coeffi- 
cient derived from it. This will give us 


ab Get emed 


dab yb 
Baha p= ple 
e—pt — 
Substituting this value of (v—a)?in the general equation of 
the circle, we have 
(c? —8°)(y—)* 
from which we obtain 


whence 


Cae oaks 


mbuadsst 
TTR 
and similarly 
CX 
oO ctR 


Substituting these values of @ and 4 in the general equation 
of the circle we have 


(see) +0-ceR) 


222 DIFFERENTIAL CALCULUS. 


whence 

x? +y? =(c+R)? 
the equation of the envelope showing it to be twofold. The 
positive value of R gives a circle with a radius equal toc+R 
circumscribing the system, and the negative value for R 
gives one that is inscribed within it. 

(123) Let there be an ellipse in which the axes vary in 
length under the condition that the area of the ellipse shall 
be constant. This condition will be expressed by the equa- 
tion 

AB=c? (1) 

To find the envelope of this curve we put its equation 

under the form 


x? 2 
Abt pet (2) 
and differentiating with respect to A and B only we have 
ci das FEE 
A’ TBs ZA 
or 
I ae a? Dea 


1A A’ eamBie vag Be 
But from equation (1) we obtain 


io lel bese 
OA Se aR 
whence 
ve 2 
Ara pot 
whence 


A=x/2 and B=yr/2 
Substituting these values in equation (1) we have 
2xy=AB=c? 
and 


_. 
ow 


C 
xy — 2 


ENVELOPES. 222 


which is the equation of a hyperbola referred to its center 
and asymptotes. The curve EF (Fig. 49) is then a hyper- 
bola, and the axis of the ellipse are its asymptotes. 

(124) Let AB (Fig. 48) and AC be the coordinate axes, 
and let the line ze of a given length move in such a manner 
that its extremities shall be at all times in the axis. What 
is the equation of the envelope described by that hne? 

Call the length of the line c, and the distance Ad and 
Aé’ respectively and a. Let Am=x and mn=y, then the 
general equation of the line will be 


ane (:) 
we have also 

a® +b" =c? (2) 
Differentiating these equations with respect to @ and éas 
variables we have 


aN a 
ia bay 
or 
y_ Ox 
eo? 
Substituting this value in equation (1) we have 
ie Gao C 
ie a? 
whence we obtain a 
a=V (2x : 
sees au 
and similarly 
b=V cry A B 


Nu e e! e'" 


ae +? =(c2x)8 + (c2y) 8c? =(c3)3 


whence 


from which : 
2 2 
which is the equation of the envelope. 


224 DIFFERENTIAL CALCULUS. 


The first differential coefficient of this equation is 


from which we learn that the curve is tangent to both 
coordinates. | 

([25) Suppose the line DC (Fig. 52) to revolve about the 
point D in the axis of abscissas, c 
varying in length so that the 
extremity C shall be at all times 


in the axis of ordinates, required & 

the envelope described by the a 

line DE perpendicular to DC at 

the point C in the axis of ordi- A D B 
nates. Fig. 52. 


Representing the distance AD by, and the tangent of 


Tp, : : 
the angle CDB by —7, its equation will be 


li 
Saa \eet) 
in whichif we make «=o we have 
nic 
ales 


tor the distance from the origin at which the line DC inter- 
sects the axis of ordinates. And since the perpendicular 
passes through the same point, its equation will be 
C 

pa PUES 5 (x) 
If we consider a in this equation as an independent variable, 
it will represent all the perpendiculars that can be drawn 
under the given condition. 

Differentiating it with respect to @ we have 


G 
Kaas —oO 
a? 


ENVELOPES. 225 


whence 


and substituting this value of @ in equation (1) we have 


é 
y = 26 


whence 

py" =46x 
which shows the envelope to be a parabola of which D is the 
focus. It also demonstrates a well known property of the 
parabola, namely, if lines be drawn from the focus perpen- 
dicular to the tangent they will intersect it on the perpen- 
dicular to the axis through the vertex. 

(126) Let AB and EO (Fig. 53) be the coordinate axes, 
and let CD be a line revolving 
between the lines AH and BK 
in such a manner that its ex- 
tremities C and D shall always 
be in those lines, and the pro-  ¢ 
duct of the distances CA and A 0 8 
DB from the axis shall be a Fig. 53. 
constant quantity. Required the equation of the envelope 
generated. 

Let OA=OB=m, and AC.BD=c?. Then producing the 
line DC until it meets the axis of abscissas at S, and mak- 
ing the tangent of BSD =a, we have 

es Bite) 2G) bt 


or 


F OF 
pe Ry ie OF 
a a 


whence 

BD=OF +am 
and similarly 

AC=OF—am 


226 DIFFERENTIAL CALCULUS. 


whence 


BD .AC=c* =OF —a*m? 
or f 
OF =a2m? +c* 
But the equation of the line CD is 
y=ax+b 
in which @ is the distance from the origin to the point where 


the line cuts the axis of ordinates, that is, the distance OF. 
Hence 


y=axt+(a*m* tec z (1) 
is the equation which, when a is variable, represents the line 
CD in every position it can assume under the given condi- 
tions. 

Differentiating with respect to a, we have 


mada 
00 ee ae O 
(a®m® + c2)2 
whence 
af 
lat mc")? 


m* 
c x 


ak 
” (m?—x?)? 


which being substituted in equation (1) gives 


C xe ae . 
dane udea es) 
(m® —x?)? 
whence 
a ‘8 
(mm? — x*)? = 762 bem 
whence 
ale 
my(m* —x*)® =c(m*? —x*) 
whence 


mm” y* = mre? —¢* x44 


ENVELOPES. 227 


or 
my? +7 x*% =m? 
which is the equation of an ellipse referred to 1s center as 
the origin, and whose semi-axes are mand ¢. 
(127) The equation of the normal line to the parabola is 


4 
y—y =F (2-2) (x) 
in which x’ and y’ are the coordinates of the point in the 
curve from which the normal is drawn, and x and y are the 
variable coordinates of the normal itself. 

If we consider x’ and y’ as variables, equation (1) will 
represent the entire system of normals which can be drawn 
to the parabola. To find this envelope of tius system we 
find the relation between x’ and y’ from the equation of the 
parabola 


J = 2px (2) 
and substitute in equation (1) the value of x’, which gives 
, 132 
AD 


whence 
2p*(y—y')=—2pxy’ +y'8 (3) 
Differentiating this equation with respect to y’ only we have 
eT mt VE BA 
whence . 
Y =V 3 pax —p*) 
Substituting this value in equation (3) we have 
pees SLs ee eg ee eye 
ap y— 2p" /2( px—p?) = — 2pxrv/ 2(pu—p?) +1 3(px—p’) |? 
whence 
1. a: 
apry +2( px—p*)(3(px—p*))* =[5(pa—p*) | 
whence 
2°y=—[8( px—p")]* =—(p)*(%—s)* 


whence 


pty = erp? (x—p) 


DIFFERENTIAL CALCULUS. 


which as we have seen (Art. 116) is the equation of the 
evolute of the parabola. 

Hence all normal lines to the parabola are tangent to the 
evolute | : 


Se BOe Lb LO) Neexs.V 3 


ArritcCAd LON Or PHEWDIFREREN TIAL CALCOLGS FO 
THE DISCUSSION OF CURVES. 


Ais CY CUO). 


({28) The cycloid is a curve described by any point in 
the circumference of a circle as it rolls along a straight line 

If for example, the circle EFD (Fig. 54) should roll along 
the straight line AB, 
the point F, starting 
frome thee point. “A, 
would describe the 
cycloid AD’B, and 
the distance from A 
to B where the gen- Fig. 54. 
erating point again meets, the line AB will be exactly equal 
to the circumference of the generating circle. 

If we place the origin at A we shall nave 

AG=«x and FG=y 

The arc FE will be equal to the line AE, and HE will be 
the versed sine of the same arc. Making DE=2r we shall 


have 


3 
FH =DH.HE=y(27—y) 


229 


230 DIFFERENTIAL CALCULUS. 


hence 
FH=GE=arc FE—x=,/p;y—y? 


whence 


x =Ver. sin. ey ee (1) 
which is the equation of the cycloid. 

The line AB is called the base of the cycloid, and ne line 
D’'E’ perpendicular to the base at its middle point is the 
axis, and is equal to 27”. , 

Since every negative value for y gives an imaginary value 
for x, the curve has no point below the base. If we make 


y=2r we have 
wit 
x=ver sin. 2r=7r 


and every value for y greater than 27 gives an imaginary 
value for x ; hence the greatest value of y is the diameter 
of the generating circle; and for all values of y between 27 
and zero there will be a real value for x. 


(129) We will now proceed, with the aid of the differen- 
tial calculus, to investigate the properties of this curve in 
reference to its tangent, subtangent, normal, subnormal, 
curvature, invclute, etc. 

Differentiating equation (1) we have 
ray riy—jty 
Vary Vane Vay (2) 
Substituting this value of @v in the general formula for the 
subtangent (Art. 52) we have 


te oe 


gra 
V ary — y® 
and for the tangent (Art. 53) 


ob 
For the subnormal (Art. 54) 
GE=¥V ary—y? 


TG= 


DISCUSSION OF CURVES. 231 


and for the normal (Art. 55) 
ier ey ee tS 
FE=y, ee —V/ 
Ve ae y® 27y 


Since GE the subnormal is equal to 4/27y —y?, which is 
equal to 4/DH.HE, the point E of the subnormal for the 
point F of the curve, must be at the intersection of the ver- 
tical diameter of the corresponding generating circle with 
the base; and the normal line =V/ 27y=1/ DE. EH must be 
a chord of that ctrcle joining these two points. 

The tangent being perpendicular to the normal will of 
course be the supplementary chord of the same circle. 
Hence to obtain the normal and tangent lines for any given 
point of the cycloid, construct the generating circle for the 

diameter D’E’ erected at the middle of the base, and 
through the given point draw the line FH’ parallel to the 
base intersecting the circle at F’. Join this point with the 
extremities of the diameter D’E’, and the line F’E’ will be 
parallel to the normal, and F’D’ will be parallel to the tan- 
gent. Hence lines parallel to these, through the given point 
will be the lines required. 

If it is required to draw a tangent parallel to a given line, 
first draw a chord from D’ parallel to the given line, and 
through the point where it meets the circumference of the 
circle draw a line parallel to the base. The intersection of 
this line with the curve of the Bs will be the point of 
tangency. 

(130) From equation (2) we hove 


dy NV ary—y' 2r 
bE ay fa 


which becomes zero Send y=2r, “ane the tangent at the 
extremity of the axis is parallel to the base. If we make 
y=o we have 


ay 


Gas oe 


232 DIFFERENTIAL CALCULUS. 


hence the tangent at the base is perpendicular to it. 
Differentiating equation (3) we have 


ordy ray 
a9 yO) Taya ee eee 
2x or eh Piet oo ve 
ie —w_y Te 
y 3 
hence 
aerial fea 
ax Pe eye 


This second differential coefficient being essentially neg- 
ative, shows that the curve is everywhere concave toward 
the base. 

(131) The formula for the radius of curvature (Art. 105) 


gives in this case, 
3 


2rax? 3 rordx?\? 
tenn Ve nats h 
ONC aes 5 axe) ; naar 
R= rax? wax se 
Be y* 


or 
R=2,/ary 

But we have found (Art. 129) the normal to be equal to 
‘/ 2ry; hence the radius of curvature at any point is equal 
to twice the normal at that point. Thus at A the radius of 
curvature is nothing, while at D’ it is equal to 2D’ E’=,;. 

(132) The equation of the evolute will be found by the 
rule given in Art. 114. 

In the equations of condition (Art. 105) 


ay 


st Foci aaa A (2) 
and 
e ax + dy" 
; a yf Bring h®, 
Substitute the values) Guam e = just found from 
ax ax* 


4 


DISCUSSION OF CURVES. 233 


the equation of its cycloid, and then, by means of that 
equation eliminate x and y. 


Thus 
ee a 
e—a=— = (y—B) 
J 
and 
dx? +-dx?(-~—1) 
baa ioe dnt yr 
yp 
whence 
y—b=2y and *—-a=—2V/ syy—y? 
or 


y=—b and x=a—2V/ —o7p—f2 
Substituting these values of x and y in equation (1) (Art. 
128), we have 
a—2/ — 27h —62 =Ver. sin.~* —b—4/— 27h—0? 
or 
a=ver. sin.~1—d-+/ — 975—52 (4) 
which is the equation of the evolute. 

(133) For all values of J that are positive a is imaginary, 
hence no part of the curve is above the base of the invo- 
lute. For all negative values of 4 greater than 27, @ is also 
imaginary, hence if we draw A’B’ (Fig. 55) parallel to the 
base at a distance 
below it equal to 
2r, the evolute 
will lie between 
that line and the 
base. Ifwe make 
b=—2r, a be- 
comes equal to 
the arc whose 
versed sine is —J, 


234 DIFFERENTIAL CALCULUS. 


that is half the circumference of the generating circle. 
Hence the point G where the evolute meets the line A’B’ is 
in the prolonged axis of the involute. If we make 
b=0, a also becomes equal to zero, and hence the evolute 
passes through the origin at A, and also the extremity of the 
base at B. For ver. sin.~!0 may be zero, or it may be a 
whole circumference. 
If we differentiate equation (4) we have 


rab rdb+bdb _— (ar-+b)adb 
/—e2rb—b? V—2rb—-B? V —27rb— 


ada= 


or 
db V —2rb—b? 


Veena 2ar+b 
showing that at the points C and B where =o the base of 


the involute is tangent to the evolute. Also since 


V/ —2rb—8? b g.."yn 
2F eam eg 
if we make 6=—27 we have 
ab 
ee 


showing the tangent to the evolute at G is perpendicular to 
the line A’B’. 


ab rey 
Squaring the value of ae and differentiating, we have 
ab® b LEK ey rv 
da® ~~ art b Mo Wa? ~~ (ar+d)? 


which is essentially negative, and since every real value of 


bis also negative, the curve is everywhere convex to the 
base of the cycloid. 

(134) These circumstances, together with the form of the 
equation of the evolute, lead us to suppose it to be an equal 
cycloid, but for certainty we will transfer the origin to G, 
and the coordinate axes to EG and GF respectively par- 


DISCUSSION OF CURVES. 235 


allel to the first. Calling the new coordinates x’ and y’ we 
have 
a=x'+m and b=)'/+n 
m and z being the coordinates of the new origin referred to 
the original axes. Then 
m=ver.sin.-12r and z=—a2r 
whence 
a=x'+ver. sin.~t27 and 6=—(2r—y’) 
Substituting these values of a and 2 in equation (4) we have 
x’+ver. sin.—127=ver. sin.—1(27—y") +4/ 2x(2r—y')—(2r— y)2 
but 
ver. sin.~!27—ver. sin.~1(27—y’)=ver. sin.—1y’ 
hence 
x =—ver. sin.~1y' +4/ o7y/ 4/2 
which is the equation of the curve CG, the values of x’ be- 
ing the same as those of x in equation (1) (Art. 128), except 
that they are negative as they should be, since the values 
of x are reckoned in a contrary direction from those of x ; 
and the curve CG is equal to the curve CF, but reversed in 
position with reference to the origin. | 

Since the curve CG is equal to FG (Art. 113) the length 
of the cycloid is equal to four times the diameter of the 
generating circle. 

(135) The character of the evolute of the cycloid may be 
demonstrated geometrically thus: 

Let us suppose two right lines AB and A’B’ (Fig. 55) to 
be drawn parallel to each other, and two circles to be des- 
cribed on the diameters DE and CE, each equal to the 
distance between the two parallel lines and tangent to each 
other at the point C. If now we suppose each circle to roll 
along the line on which it stands, at the same rate, so that 
they are at all times tangent to each other, then the point 
C of the upper circle will describe the first half of a cycloid 
CPF, while the same point C of the lower circle will des- 
cribe the last half of an equal cycloid CP’G. 


236 DIFFERENTIAL CALCULUS. 


Suppose the two circles to have arrived at the point C’ in 
the line AB, and that P is a point in the upper curve. The 
diameter DC of the upper circle will have assumed the 
position PO, and the diameter CE of the lower circle will 
have assumed the position O'P’ parallel to it; and P’ will 
be the generating point of the lower cycloid. 

Draw the chord PC’ and it will be normal to the upper 
cycloid (Art. 129). Draw also the chord C’P’, and it will 
be tangent to the lower cycloid at the point P’ (Art. 129). 
Now since PO and O’P’ are parallel, these two chords and 
the corresponding arcs are equal, and hence the angles 
PC’D’ and P’C’E’ are equal; and since D’E’ is a straight 
line P’C’P is a straight line also, normal to the upper curve 
and tangent to the lower one. Hence the lower cycloid is 
the evolute of the upper one. 

(136) The equation of the evolute may also be obtained 
by considering it as the envelope of the normals drawn to 
the curve. 

The general equation of the normal to the cycloid is 


f 


Ne ectivine 5 1O (1) 
in which «’ and y’ are the coordinates of that point of the 
cycloid to which the normal is drawn; and x and y the gen- 
eral coordinates of the normal line. If we make x’ andy’ 
variables, still retaining their relative values, as in the equa- 
tion of the cycloid, the equation (1) will represent the whole 
system of normals that can be drawn tothe curve. Ifnowwe 
eliminate one and make the differentials of the equation 
with respect to the other equal to zero, then (Art. 120) by 
eliminating that we shall have an equation which will be 
that of the envelope of the normals, and also the evolute of 
the cycloid. 

‘Substituting for x’ in equation (1) its value taken from 
the equation of the curve (Art. 128) we have 


DISCUSSION OF CURVES. 237 


t 


y— y= ea ——= > (w— ver. sin.—1y/+V ary’ —y'*) 
V ary’ — yf? 


or 
eae L-VVCL: Slee yn 
bs aN sag aaa ee 
whence 
yV ary’ — y'®—y'ver. sin.—1y’ +y' =o (2) 
Differentiating this equation with respect to y’ we have 
r—y 


fs 
ee Vs sin. cork a 
2 AY Spee ae: Sime? V ary! —y* 
Substituting in this equation for ver. sin.—1y’ its value taken 
from the equation of the cycloid, and multivlying by 
V ary’ —y/*, we have 
V(r) (2! +V Say aryl ag THAW aryl yl =O 
or 
Wt HV aryl =yl® tary —y"* ty 1 —2V/ ary! —y/? 
or 


VW r—y)=—(x—2')V/ ary’ —y® +377 waist 
but 


peer 
x—x/ = 


hence 


Wr—y")= ae Claas Oo Mehta ens f 

clearing of fractions and multiplying we have 
i VY E27 yy —27y *— yee a eanyeey, ® 
whence 
Ty aioe (Sg Of Vay 
Substituting this value of y’ in equation (2) we have 

yV —ary—y? +y ver. sin.-1—y—xy=o 

or 


x«=Vver. sin.~! —y+V —2ry— y? 


238 DIFFERENTIAL CALCULUS. 


which is the equation of the envelope of the normals, and 
also of the evolute of the cycloid, as in Art. 132; for sub- 
stituting the variables @ and 4 for x and y, the equations are 
identical. 


THE LOGARITHMIC CURVE. 


(137) The logarithmic 
curve is one in which one 
of the coordinates is the 
logarithm of the other. 

Its equation is 

x=Log. y 
If we represent the base of 
the system by a the equa- 9 ) | 
tion may be written =5- 4-322-JuA Places 
y=at Fig. 56. 

The curve may be constructed by laying off on AB (Fig. 
56) the axis of logarithms, the numbers 1, 2, 3, 4, etc., on 
both sides of the origin, and laying off on the corresponding 
ordinates, or on AC the axis of numbers, the corresponding 
powers of a. | 

When x=o then y=1, whajever may be the value of a, 
and hence all logarithmic curves will intersect the axis of 
numbers at a distance from the origin equal to 1. 

If ais greater than 1, and x positive, y will increase as x 
increases, and there will be a real value of y for every value 
of x as in the curve DE. 

If x is negative, then the value of y is fractional, and 
decreases as x increases negatively, but y will not become 
zero until x=— o. 

If y is negative, there is no corresponding value of x, and 
hence the curve can never pass below the axis AB. 

If ais less than 1, then y will diminish as x increases 


DISCUSSION OF CURVES. * 239 


positively, and becomes zero when x=; but y increases 
for negative values of x, and the curve has a position the 
reverse of the first as DD” in the figure. 

(138) If we differentiate the equation 


Var 
we have 
1h I I 
aN avy. a = y— 
ax m1 m 
and 
aay I I 
me ee Ba) aa 
Axe m*” m7 
ay 


If we make y=o we have pha hence the tangent for 


that value of y is the axis of abscissas; and since yo gives 


x—=— o the axis of abscissas is an asymptote to the curve 
: ; a. 
(Art. 88). But since y=oo gives x= 0, and also ee 00, 


the curve has no tangent parallel to the axis of ordinates 
except at an infinite distance. The sgn of the second dif- 
ferential coefficient shows that the curve is at all times con- 
vex toward the axis of abscissas. 

The subtangent PT =F y7=M; hence the subtangent is 
constant and equal to the modulus of the system of logar- 
ithms, to which the curve belongs. 

In the Naperian system the modulus is 1, and in this case 
PT and DA are equal. 

(139) We will now investigate the curve whose equation is 

y=x log. x 

Every value for x gives a single value for y. 

If x is less than 1 the value of y is negative. (1) 

If x is greater than 1, y is positive. (2) 

lige Ome 4-21) —0: (3) 

If x is negative, y is imaginary. (4) 


240 DIFFERENTIAL CALCULUS. - 


If we differentiate the equation we have 


ay 
Te OS +1 (5) 
zi py eT 
ax" — x (6) 
a 
Making ae we have 
I I 
— —— —1 = = 
log. x TSOL =F San ae (7) 


which corresponds to a minimum as shown by the positive 


9 
tates | 


J 
value of Tee” (3) 
ae 
When «=o, ie tea (9) 
4) 

When «=1, 7 =1. (10) 

Since y is negative between x=o and x=1 and then pos- 
fs Lid ait i ne 
itive, while gt is always positive, the curve 1s concave 
toward the axis of abscissas between «=o and «=1, and 
afterwards convex. (11) 


Hence the curve begins at the 
origin (Fig. 57) and intersects the 
axis of abscissas at D, making 
AD=1 (3).- The tangent to the 
curve at D makes an angle of 45° 
with the axis of abscissas (10), 
while at A the axis of ordinates is 
tangent (9g). Atthe point E, whose 


abscissa is , the tangent to the curve is parallel to 


I 
2e7 Loz 
the axis of abscissas (7), and the ordinate is at a minimum 
(or negative maximum) (8). Between A and D the curve is 
below the axis of abscissas (1), and concave to it (11); and 


DISCUSSION OF CURVES. 241 


beyond D the curve lies entirely above the axis of abscissas 


: / : I : 
and is convex toit. Sincex= PPR PGS CoS a have 
FE=AF 
(140) We will next take the equation 
1 
ee ae oT 
Damian Ener 
a 


Every value of x gives a real positive value for y, and 


hence there can be no negative value of y. (x) 
If =o, y=o, and hence the curve passes through the 
origin. (2) 
If x is negative we have y=e”, in which if «=o, y will 
become infinite. (3) 
So that x=o gives two values for y, according as x ap- 
proaches zero from the positive or negative side. (4) 


If « be negative and increase in value, that of y will 
approach more nearly to 1, which it will reach when 


H=— Or, (5) 
If x be positive, and increasing the value of y approaches 
more nearly to 1, which it reaches when x= ©. (6) 


Hence the curve will be as in 
Fig. 58, in which AB and AC are 
the axes of coordinates, and DE a 
line parallel to AB ata distance 
from it equal to 1. It will pass 
through the origin A (2), extend 
indefinitely in a positive and nega- 
tive direction, and the line DE will Fig. 58. 
be an asymptote to both branches (5), (6). The axis of 
ordinates will also be an asymptote in the positive direction 
(3) (Art. 88). As the branch DC of the curve extends to an 
infinite distance in both directions, it has no connection with 


242 DIFFERENTIAL CALCULUS. 


the branch AE, which commences at the origin and is infi- 
nite at the other extremity. There are, in fact, two curves, 
one answering to the positive, and the other to the negative 


value of x. 
1 


If we differentiate the equation y=e * we have 
1 


MeN AP ia 
ax x 
and 
_1 
aPy Se (tT —22) 
ax ee 
Since 
ant’ 
e * AT 
a atas r 
wer 
we shall have 
ian 
oa a 


when either x=o or x=, hence the axis of abscissas is 
tangent at the origin, and parallel to the tangent at an infi- 


nite distance in either direction; in which case y=r1 (5) (6). 
For all negative values of x, = is positive, and hence 
(oe 


the branch DC is convex to the axis of abscissas. For all 
fe Cy 45 
positive values of x less than $, a is also a positive, show- 


ing that between A and H the curve is convex to the axis 
of abscissas, while at the point H, where x=4, the value of 
ae a ; : 

a changes from positive to negative passing through zero, 
showing that at P the curve ceases to be convex, and be- 
comes concave toward the axis of abscissas. ‘This is called 
an inflexion. 


SECTION XVI. 


winwiat Ane eOLVe LS. 


(141) Singular points of a curve are those at which there 
exists some remarkable property not common to other points 
of it. Such, for example, as the maximum or minimum 
value of the ordinates or abscissas, points of inflexion, con- 
jugate points, cusps, etc. 

In many cases these points are easily discovered by the 
aid of the differential calculus, as will be seen by the fol- 
lowing examples. 


MAXIMA AND MINIMA. 


(142) If we differentiate the equation 


=3+2(*—4)* 
we shall have 
2 =8(x—4)9 
Tf a4(0—4)? 
se =48(«—4) 
TJ _ 48 


Here we find that x=, will reduce the first differential 
243 


244 DIFFERENTIAL CALCULUS. 


coefficient to zero, showing that the tangent to the curve Is 
parallel to the axis of abscissas (Art. 36),and hence the 
value of the ordinate way be amaximum or minimum. But 
since the second differential coefficient is always positive 
except when it is zero, the first must be an 
increasing function, and hence at zero must p 
be passing from negative to positive, and the 
value of y must be changing from a diminish- — 
ing to an increasing one. So that there is a 
minimum when x«=4, as shown in Fig. 59. 
We infer the same thing from the sign of the fourth differen- 
tial coefficient (Art. 209). 

If we take the equation 


C B 


Fig. 59. 


y=2—2(x—2)4 
we shall have 


a 

me aad 
(hae 

T} = o4(a—2)? 
ae y 

ed —48(x—2) 
d4 

axt 2 


Since x=2 reduces the first differential coefficient to zero 
the tangent at that point is parallel to the axis 9 


of abscissas; and since the fourth differential ae 
coefficient (the first that has real value for 

x=2) is negative, the value of y at that point cane 
must be a maximum as in the figure. Since Fig. as 


9 
w 


De ; : 
Tue 1s at all times negative, except when x=z2, the curve 


will be concave toward the axis of abscissas for all positive 
values of y (Art. gr). 


a 


SINGULAR POINTS. 245 


POINTS OF INFLEXION. 


(143) A point of inflexion is one in which the radius of 
curvature changes from one side to the other of the curve 
so that it will be convex on one side of the point of inflex- 
ion, and concave on the other towards any line not passing 
through the point itself, and this will, of course, be true for 
the axis of abscissas, and hence at such a point the second ° 
differential coefficient will change its sign. If the point of 
inflexion should be in the axis of abscissas, both parts of 
the curve would be convex or concave to it, but the second 
differential coefficient will still change its sign (Art. 93). 
Now in order that the sign should be changed, the function 
must pass through zero or infinity, and hence the equations 

as ary 

yiiae and eee Cs 
will give all the points of inflexion in any curve in which 
there may be such points. 

(144) Let us now take the equation 

pene 


whence 

ay 

Te Ne 2)? 
and 

if 2 

<5 =18(2—2) 

ay 

eee 


In this case every value of y gives one for x, and wice 
versa, hence the curve has no limit. When x=a2, then 


a 
Po — and y=1, so that if wemake AC=2 


and CD=1 (Fig. 61), the tangent at D will 

be parallel to the axis AB. But x=2 re- 
a” 

duces se to zero, also indicating that 


246 DIFFERENTIAL CALCULUS. 


there may be a point of inflexion, hence we resort to the 
ge ree et: : 
value of =e which is a positive constant. From this we 


Sele dos . ; 
learn that at zero yae iS an increasing function, hence it 


must pass from negative to positive, showing that at the 


eee, ; 
same point x Passes from a decreasing to an increasing 


function, and hence does not change its sign, but remains 
positive both before and after the zero point; and this shows 
that the value of y is an increasing function both before and 
after the same point. There is, therefore, no maximum nor 
minimum for it at that point. 


9 
@ 


y ee = 
Since ae changes its sign at x=2 from negative to pos- 


itive, the curve will be concave toward the axis of abscissas 
when «<2, and convex when x>2, so that at the point D 
where «=z the curvature changes its direction and there is 
an inflexion. 

(145) If we take the same equation, and make the last 
term negative, we shall have 


ay “if } D 
ax =—9(x—2) 
a 29 
—_ ie A C B 
ax* 18(x 2) Fig. 62. 


The point D where x=2 will still be the point of in- 
ay . 

flexion, but since <4 is negative for all values of x 
except «=2, the curve will approach the axis of abscissas 
me ; Aa 
for all positive values of y, except y=1, and since ey 
is positive for r<2, and negative for x>2, the curve will be 
‘convex toward AB between A andC, and concave afterwards, 

as in Fig. 62. 


SINGULAR POINTS. 247 


The first differential coefficient being zero when «=z, it 
follows that the tangent at that point will be parallel to the 
axis of abscissas (Art. 36), and hence the curve will pass 
fromone side of the tangent to the other at the point of tan- 
gency, and will be convex to the tangent on both sides of it. 

(146) If we take the equation 


petal =2)e 


we have 

w_ 6 
s(x—2)8 
dey T2 

—— 7 
ie 25(x—2)* 

If we make x=2 we have 
dy @? 


efile Cm oe 2 aa 
and hence at the point D where «=2 the tangent will be 
perpendicular to the axis of abscissas (Fig. 63), ‘ 
and since s is positive for other values of x, 


the curve will leave the axis of abscissas, for all j——¢__g 
positive values of y as x increases. And since Fig. 63. 
oe ant a. Da hes : 
=e changes its sign from positive to negative 1n passing 
through infinity where «=z, the curve will be convex toward 
the axis of abscissas for «<2, and concave for «> 2, and at 
x—=2 there will be an inflexion. 
(147) If in the same equation we make the last term 


negative we have - 3 
y=2—2(x—2)? 


and 
YY : 
we 5(x—2)8 
Vay) or 
Mt 


248 DIFFERENTIAL CALCULUS. 


and the conditions will be changed so that the curve will be 
reversed. It will now approach the axis of 
abscissas, and the second differential coeffi- 
cient will change its sign from negative to 
positive in passing through infinity where 
x=2; the curve will be concave for x<a2, A 
and convex for «> 2. 

The point D (Fig. 64) will still be a point of inflexion, 
and the tangent will be perpendicular to AB. 

(148) If we take the equation 


Fig. 64. 


y=(x—2) 
we have 

ay 

Wn a 3he—2)* 
and 

wey 

Wee =6(x—2) 


which all reduce to zero when x=2. 

This shows that the curve meets the axis of abscissas at 
the point where x=2, and that this axis is 
tangent to it there. And since the second 
differential coefficient will have the same 
sign as y (both being the same as that of 
x—z2), it will change from negative to posi- Fig. 65. 
tive at the point where «=z, showing an inflexion there, and 
that the curve is convex to the axis of abscissas on both 
sides of it. 


Cc B 


CUSPS. 


(148) A cusp is a curve consisting of two branches start- 
ing from a common point in the same direction, and imme- 
diately diverging from a common tangent. They are of two 
kinds, namely: Those in which the branches are on differ- 


SINGULAR POINTS. 249 


ent sides of the tangent, which are cusps of the first order; 
and those in which the branches are both on the same side 
of the tangent, which are cusps of the second order. 

The following are examples of the first order. 

Let 


y=1+3(e-1)8 


then 
Oa a(x—1) 4 
and 
CDR 2 
@)* x(2—3)8 


dy J 
ee ae Ten and Pee mek 


For every value of «<1, es is negative, and positive for 


every value greater. The curve, therefore, approaches the 
axis of abscissas, in the first case, and recedes from it in the 
second (y being positive), which indicates a minimum, while 
the tangent at that point is perpendicular to 


! ekiean ye D 
the axis of abscissas; and since ee always 
negative the curve is concave toward thesame A CB 


Every value of 7 less than 1 gives an imaginary value for 
x, while every value greater than 1 gives two values for x, 
one less and one as much greater than 1. Hence the curve 
has two equal branches commencing at D (Fig. 66), where 
they have a common tangent. 

(150) If we make the last term negative, the signs of 
dy 


eae k 
ea and Sa will be reversed, and the first will change from 


250 DIFFERENTIAL CALCULUS. 


positive to negative as x passes from x<1 to 


x>1; while at x=1 the tangent is still perpen- D 

dicular to the axis of abscissas. Any value ofy | __ AN 

greater than 1 will give an imaginary value for Al C R 
Fig. 67. 


x, while every value less than 1 will give two 
real values for x equally distant from the point C where 


9 
w 


a 
x=1 (Fig. 67). The sign of a being now always posi- 


tive (except at x=1) shows that both branches of the curve 
are convex toward the axis of abscissas. ‘These are then 
cusps of the first order. 
(151) If we differentiate the equation 
y=ok(e—1)h (x) 
we have 
ay 1 
aoe +3(a—1)? 
gy 
an 
We see from equation (1) that when x=1, y=2, and when 
“<1, y is imaginary, while when «>1, y has two values, one 
greater than 2 and the other as much less; so that DC (Fig. 
68) be drawn perpendicular to AB, making DC=2, the curve 
will commence at D and be symmetrical about the line DE, 


=+23(x—1)# 


Z} 
and since = =o for the point D, the line DE will be tangent 


to both branches. Since for every other value of 
dy : i 
Xn has one negative and one equal positive 
value, one branch of the curve will approach the A ¢ B 
axis of abscissas, and the other recede from itat Fis. 68. 
2 
° ra 
an equal rate. And since for every value of x> Sears 
has two equal values with contrary signs, the positive cor- 
responding with the greatest value of y, we infer that the 


SINGULAR POINTS. 251 


upper branch of the curve is convex, and the lower branch 
concave, to the axis of abscissas, and that the curve is a 
cusp of the first order. 

({52) If we change the sign of the last term and make 
the equation a 
yH24(1—x)* 
we have 


1 
2 = +3(1 —x)* 
A i a 
2 i 
aa) 
and the curve will be similar, but reversed in position as in 
Fig. 69. | 
If «>1, y will be imaginary. D 
If x=o, y=3 and y=1. 


9 
C4 


EY 
If y=o, x=1—4/4. Since sis both pos- A} C 8B 


itive and negative when «<1 there is no maxi- Fig. 69. 
imum nor minimum value for y. 
(153) The curve represented by the equation 
(y—2? P= x? 
contains a cusp of the second order, as well as some other 
singular properties. 
Solving this equation we have 


5 
you tx* (1) 
and by differentiation we have 
dy 3 
—= 5 yk 
We ee Ee 


and 
A A 
an” 4 


From equation (1) we find that the curve passes through 


252 DIFFERENTIAL CALCULUS: 


the origin, and does not extend to the 


negative side of the axis of ordinates. / 
Every positive value for «<1 gives two 

real positive values for y, while x=1 | Dp 

gives one positive value for y and one A | ZE Se 
equal to zero. Hence the curve has E C¢ 

two branches, both of which pass packs 


through the origin, and one intersects the axis of abscissas 
at a distance from the origin equal to 1. 


wy 
If we make Pe ede have x=o and x=3%. Hence 


there are two points in which the tangent to the curve is 
parallel to the axis of abscissas; at the origin where the 
axis itself is tangent and at the point D (Fig. 70) whose 
abscissa is «=$$; and as the value of x at this point 
derived from the equation aoe corresponds to the minus 
sign in equation (1), the point of tangency is on the lower 
branch of the curve. 


: : . Loa 
The second differential coefficient has two values 2-2 xt 


wk 
and pee hed of which the first belongs to the upper branch 
4 


of the curve and is always positive, while the second is pos- 


: Wee eer ; : 
itive so long as a is less than 2; that is so long as x is 


d* : 
less than 34, which makes Sa te After that it becomes 


negative; showing that the lower branch of the curve is 
convex to the axis of abscissas, as far as the point whose 
abscissa is AE=°4,, and at this point there is an inflexion, 
the curve becoming concave to the axis of abscissas as long 
as y is positive and convex afterward. Hence at the origin 
there is a cusp of the second species. 


SINGULAR POINTS. 253 


CONJUGATE POINTS. 


(154) Conjugate points are those single points which are 
isolated from the curve, but will satisfy the equation. — 
If we differentiate the equation 
a a? (x —5) 
yar, /t1e—4) 
VA a (1) 


we have 
Ly 3x20 
ax 2 a(x—b) 
aey 3x— 4b 
dx? = 


If we make x=o in equation (1) we have y=o, but any 
other value of x less than 4 will make y imaginary. Hence 
while the origin will satisfy the equation, that point is iso- 
lated, having no connection with the curve. We also see 
that «=o will give 

ay b 

Tia al 
which is imaginary as it should be, since at that point the 
curve can have no tangent. 

If we make «=4, we have 


showing that the tangent at that point is perpendicular to 
the axis of abscissas, while the value of 
y is zero. As every positive value of 
x>6 gives two equal values for y with 
opposite signs, the curve is symmetrical 
about the axis of abscissas, and as the 


dL 
value of os has the same sign as y, the 


Fig. 71. 


curve departs from that axis in bothdirections. If we make 
x negative the value of y becomes imaginary ; showing that 


254 DIFFERENTIAL CALCULUS. 


the curve does not extend to the negative side of the axis of 


ordinates. 
Ley 
lf we make ——>=0, we have 
ax? ? 


46 

3 

showing that at the points C and C’ (Fig. 71) which lie in 
the ordinate drawn through D at a distance from the origin 
equal to 42, the curve has an inflexion in each branch, since 
for that value of « we have 


x— 


Ah 
aus 
s 


If we make x<—,, the second differential coefficient will 
°) 


46 
have a sign contrary to thatof y. If ae the signs will 


be thesame. Hence between H and D the curve is concave 
toward the axis of abscissas, and convex beyond D, which 
also shows an inflexion. 


a 
If we make seas we have 3x==24, or 


2b 
CS 


3 
This value being substituted for x in equation (1) gives 


an imaginary value for y, showing that there is no point in 
the curve where the tangent is parallel to the axis of abscissas. 


MULTIPLE POINTS. 


(155) A multiple point is one in which two or more 
branches of a curve intersect each other. At such a point 
the curve will always have as many tangents as there are 


why 
branches, and hence Ye Must have the same number of val- 


ues for that point. 


SINGULAR POINTS. 255 


Let us take the equation 
y=bt(x—a)\/x—c where a>c (1) 
then by differentiating we have 
ay ae 

WRC ha Fe a fee 

For «=a and x=c in equation (1) we have y=d ; hence 
H and H’ (Fig. 72) corresponding to 
these values of x and y are points in 
the curve. For all values of *<c 
that of y is imaginary; hence there 
is no part of the curve between H and A 
the axis of ordinates. For every 
value of x>c, except x=a, y has two Fig. 72. 
values, one greater, and the other as much less than @. 
Hence the curve is symmetrical about HH’. For x«=c we 


ad : : 
have = oo, hence the tangent at H is perpendicular to 


. . J 
the axis of abscissas. For «=a we have two values of 3 
or 


equal to each other with contrary signs, namely, «1/2 —c and 
—/x—c. Hence at H’ there are two tangents making sup- 
plementary angles with the axis of abscissas, so that the two 
branches of the curve cross each other at that point in direc- 


: dy 
tions symmetrical with HH’. If we make Te =o We have 


a-+2¢ 
3 
which shows that at the point corresponding with the ordi- 
nate at E where AE equals one-third of (2AC+AB), the tan- 


“a= 
OV =—— 


(156) We will close the discussion of algebraic curves 
with that of the equation 
ay® —x° +(b—c)x* +bcx=0 
Solving this equation with reference to y we have 


256 DIFFERENTIAL CALCULUS. 


5 eee 
a (1) 
and 
ay, 3x*—2n(b—c)—be 
dx ~~ 2N/ ax(x—b)(x+c) 

If in equation (1) we make x=o0, x=4, or x=—<c, we have 
in évery case y=o. Hence there are three points, H, A and 
H’ (Fig. 73), where the curves meet the axis of abscissas. 

Every negative value of «>c gives an imaginary value 
for y, hence the curve has no point on the negative side of 
H, since AH=c. Every negative value of «<c will give 
two equal values for y with opposite signs; hence from H 


to A the curve is symmetrical about the axis of abscissas. 


Every positive value for «<4 gives an ,; 
imaginary value for y,; hence no part 
of the curve lies between A and H’. y iis B 
Every positive value for x>b gives 
two equal values for y with contrary 
signs. Hence on the positive side of 
H’ the curve is symmetrical about Fig. 73. 
the axis of abscissas, and the entire curve consists of two 
parts having no connection with each other by a common 
point. 

Each of the values of x that reduce y to zero also reduce 


a ere: : 
= to infinity; hence at the points H, A and H’ the tangent 


is perpendicular to the axis of abscissas, and one of these 
tangents is the axis of ordinates. 
If we solve the equation 
3x? — 2x(b—c)—bc=o 
we shall have 
ltt V 3bc+(b=0)? 
3 
but /3dc+(d—c)*<é+c,; hence if we take the positive 


SINGULAR POINTS. 257 


value of the radical part the result will be less than 
b—e-+b--e ; he tee ; 
——.——, that is, less than $4, hence it will give no point 
of the curve. If we take the negative value, the result will 
be numerically less than —%c,; hence there will be two 
points where the tangent will be parallel to the axis of 
abscissas, corresponding to the point on that axis where 


_b=—c—V 3bc+(b—c)? 


3 
If c=o the equation becomes 


XxX 


ay*® =x3 —bx* 
in which case the oval HA is contracted into a conjugate 
point at A as in Art. 154. 
If d=o the equation becomes 


ay* =x> +cx? 


x ® tex" 
y= 4£\/— 


Ay 3k eee 

dx ~~ 2/ gx?(x¢-+4c) 

In this case the curve takes the form 
in Fig. 74. There are two equal values 
for y with opposite signs for every value 
of x on the positive side of H where 


or 


and 


mea 
x=—c. Atthat point =~=o, and the 


ax 
tangent is perpendicular to the axis of Bie te: 
: ay 2¢ 
abscissas. If we make Vin ats have x=o and a ae 
ac 
hence the tangent at A and at T and T’ where eee are 


parallel to the axis of abscissas. 


258 DIFFERENTIAL CALCULUS, 


If we make both 4 and ¢ equal to zero we have 


whence 


and 


In this case the curve assumes the 
form in Fig. 75. There is no negative 
value for x, and all positive values of x 
give two equal values fory with contrary 
A, wv 
signs. At the origin we have ee and 
hence the axis of abscissas is tangent to 


Fig. 75. 


both branches of the curve which is a cusp of the first 


species 


PART II. 


oe 


INTEGRAL CALCULUS. 


INTEGRAL CaALcuLus. 


SECTION I. 


PRINCIPLES OF INTEGRA TION. 


(157) The problem of the differential calculus is to 
obtain the differential or rate of change in a function arising 
from that of the variable, or variables, which enter into it. 
The corresponding problem of the integral calculus is to 
pass from a given differential of a function to the function 
- itself, 

The first of these operations can always be performed 
directly by rules founded on philosophical principles. The 
second can only be performed by empirical rules founded on 
actual experiment. We cannot proceed a@recdly from the dif- 
ferential to the function, but, as it were, backwards; that is, 
we show that a function is the integral of a given differen- 
tial by showing that the latter would be produced'by differ- 
entiating the former. Thus we know that x? is the integral 
of 2xdx, because 2xd¢x has been shown to be the differential 
of «*, Hence the rules for integration are merely the rules 
for differentiation inverted. 

While rules have been obtained for differentiating every 
algebraic function, it by no means follows that every differ- 


a 261 


= 


262 INTEGRAL CALCULUS. 


ential can be integrated. The number of simple algebraic 
functions is very small, and each one has its specific form 
of differential. Should any function be complicated, it can 
be analyzed and differentiated in detail, applying only the 
rules for simple forms. But before a differential can be 
integrated, it must be reduced to one of the forms arising 
from differentiating a simple function; and this can be done 
in comparatively few of the infinite number of forms that 
differentials may assume. The transformations available for 
this purpose form one of the chief subjects that demand the 
attention of the student of the integral calculus. The dif: 
ficulty of integration is very much increased when the dif: 
ferential is a function of two independent variables, for the 
rate of change in such a function can give but little indica. 
tion, generally, what the function is. 

There is still another difficulty in obtaining the exact 
integral of any given differential. We have seen that the 
constant terms in any function disappear when it is differ- 
entiated, and, of course, when we come to integrate an iso- 
lated differential expression, we cannot know what constants, 
if any, should belong to it. Insuch a case, then, we pay no 
attention to the question of constants. If. however, the 
function should occur in an eguation we can generally find 
from the conditions expressed by it what value would belong 
to the constant. Until this is done we indicate by adding 
the symbol.C to the integral that a constant is to be supplied 
if needed to render the integral definzte. Until then it is 
sald to be zudefintte. 

The notation indicating the integral of any differential is 
the letter s elongated, thus /xd@x would be read “the integral 
of xdx.” This notation was originally adopted by Leibnitz 
to indicate the sz of the infinitely small differentials or dif- 
ferences of which he supposed the function to be made up, 
and is still retained as a matter of convenience even by 


PRINCIPLES OF INTEGRATION. 265 


those who reject its original meaning, as employed in the 
system of Leibnitz. 

The following rules for integration are derived from those 
for differentiating ; being in fact the same rules inverted. 

(158) Lf the differential have a constant coefficient it may be 
placed without the sign of integration. 

For we have seen (Art. 10) that the differential of a vari- 
able having a constant coefficient is equal to the constant 
multiplied by the differential of the variable; that is to 
say, the coefficient of the variable will also be the coeffi- 
cient of its differential; hence the coefficient of the differ- 
ential will also be the coefficient of its integral, that is, of 
the variable; and may be placed outside the sign of integ- 
ration. 

Thus 

dax)=adx hence fadx=afdx=ax 

(159) Zhe integral of a differential function, consisting of 
any number of terms connected together by the signs plus and 
minus, 1s equal to the algebraic sum of the integrals of the terms 
taken separately. 

For we found (Art. 9) that the differential of a polynom- 
ial is found by differentiating each term separately, hence to 
return from the differential to the polynomial, which is the 
integral, we must integrate each term separately. Thus 

Ax+y—sz)=dx+dy—dz 
hence | 
S (dx t+dy—adz) =fdext/fady—fdz=x+y—2 

(160) Zhe integral of a monomial differential consisting of a 
variable, multiplied by the differential of the variable ts equal to 
the variable raised to a power with an exponent increased by one, 
and divided by the increased exponent and the differential of the 
variable. 

We have in (Art. 15) the rule for obtaining the differen- 
tial of the power of a variable. In other words we have 


264 INTEGRAL CALCULUS. 


given the steps by which we pass from the power to its dif- 
ferential; and hence to pass back from the differential to its 
integral, that is, the power, we must retrace each step. Thus 
in the first case we diminish the exponent by one; in the 
latter we increase it by one. In the former we multiply by 
the differential of the variable; in the latter we divide by 
it. In the former we multiply by the exponent before reduc- 
ing it; in the latter we divide by the exponent after increas- 
ing it. Thus 

DEBE heer 
because 

ax” =ux" "qx 

(161) Ifthe function consist of the power of a polynom- 
ial multiplied by its differential, the same rule will apply. 
Thus let the differential be 
(ax+x*)"(at2x)dx=(ax+x*)"dax+x?) 


make 
ax+x* =u 
then 
(ax+x?)" (at2x)dx=u" du 
and é 
ynrti (ax+x?)r+1 
Si du= a PEE ey 
HAS So ie 
EXAMPLES. 
i : x? ax x 
Lx. 1. What is the integral of : ? Ans. FF 
| 4 
, ; ak ax 
Ex, 2. Whatis the integraliofix*adxr « Ans. i 


ax 
Ex. 3. What is the integral of yee Ans, 2/ x 


; : ax : I 
Lx. 4. What is the integral of act Ans, oS eee 


PRINCIPLES OF INTEGRATION. 265 


, : hy 
Ex. 5. What is the integral of ax?dx+ nye? 
ax? 7 
Ans. 3 + x 


(162) If the exponent of the variable in the case pro- 
vided for in Art. 160 should be —1, the rule will not apply. 
For by this rule 

ey 

emt ee 00 

rs Seats 
and this arises from the fact that a differential with such an 
exponent can never occur under the rule given in Art. 15; 
for then the variable must have been x°, a constant quan- 
tity, that cannot be differentiated. Such differentials, how- 
ever, do frequently occur, but the rule for their integration 
must be drawn from a different source. We have found 


: : ax 
(Art. 38) that the differential of log. a oe c-l7x~, and 


hence a differential of this kind must be integrated by the 
rule derived from that given for differentiating logarithms. 
That is to say, the integral of any fraction, in which the 
numerator is the differential of the denominator, is the 
Naperian logarithm of the denominator. 


EXAMPLES, 
; ' adx 
£x.1. What is the integral of —\~? Ans. alog.x 
$ ; 2bxdx 
Ex, 2. What is the integral of rary a: ? 
Ans. log. (a+édx?) 
: ; adx Qt. 
Ex. 3. What is the integral of |? Ans. log. x 
An” oie 
Ex. 4. What is the integral of 3 ? Ans. alog. x 


ax 
/x.5. What is the integral of perce? Ans. log. (a+<x) 


266 INTEGRAL CALCULUS. 


(163) 2 the differential be in the form of a polynomial, 
raised to a power denoted by a positive integral exponent and 
multiplied by the differential of the variable, the integral may be 
found by expanding the power and multiplying each term by the 
differential of the variable. We may then integrate the terms 
separately. ‘Thus let us take the expression 

(a+bx)* dx 
Expanding the binomial and multiplying each term by dx 


we have 
ardxt+2abxdx +b? x* dx 
which may be integrated as in Art. 158. 


EXAMPLES. 


Ex. 1. What is the integral of (5 -+7x?)?dx ? 
: Ans, 25x+-4be? +S hxb 

Ex. 2. What is the integral of (a+3x?)® dx? 
Ans. a®x+3a*x3 +2 hax) 42,1417 
(164) 7 a binomial differential be of such a form that the 
exponent of the variable without the parenthesis ts one less than 
that of the variable within, the integral will be found by increas- 
ing the exponent of the binomial by one and dividing it by the new 
exponent into the exponent of the variable within into tts coefficient. 

For suppose the differential to be 
(atdx” Marlex 


make 
atbx”" =p 
then 
ap=nbx"-ldx 
and 


ap 


Bal Pee, 
x” 1ax oF 


from which 


mM J, 
(atdx )\x"—-lax — ue 


PRINCIPLES OF INTEGRATION, 267 


of which the integral is 
prt (atdx” \m+t 


(m+1)nb~ (m+1)nb 
hence the rule. 


EXAMPLES. 
1 
Ex. 1. What is the integral of (a+dx*)* mxdx? 
m ee 
Ans. at + bx?) 


Lx. 2. What is the integral of (a? x%) Bde? 
Ans. (a? 42)2 
Ex. 3. What is the integral of (a-bbx") Pex? 
cla + bx2)2 
sae 
({65) Every rational fraction, which is the differential of 
a function of «, may be put under the form 
Ax®™ +BaM14Cym-2+ 1... Dx+E 
Fat +Ger-1 tHe Kai” 
in which the greatest exponent of the variable in the denom- 
inator exceeds by one or more the greatest exponent in the 
numerator. For if it is equal or less, a division may be 
made, until the exponent of the remainder would become 
less than that of the divisor, and this remainder would be- 
come the numerator of the fractional part of the quotient; 
the other part, consisting of entire terms, would be integrated 
as in Art. 159. Hence we need only to consider the method 
of integrating the fractional part of the quotient, or rather 
any fractions of the form already given. 
(166) For this purpose we resolve the denominator into 
factors of the first degree at 
(x—a)(x—b)(x—c)(x—Z2Z) ete. 
and place the fraction under the form 
( A B Cc 


D 
x—a a aa etc. ax 


Ans. 


268 INTEGRAL CALCULUS. 


in which A, B, C, D, etc., are constants, whose values are 
determined by reducing all the fractions to a common de- 
nominator, and placing the sum of the numerators equal to 
the original numerator (the denominators being identical). 
Since this equality of the numerators must exist, Indepen- 
dent of any particular value of x, the coefficients of the like 
powers of x must be respectively equal to each other; and 
this will furnish enough equations to determine the values 
of the constants. Substituting these values the fractions 
may then be integrated separately. 
(167) For example let us take the fraction 
2aax 
xe ane (x) 
which by decomposing the denominator may be put into the 
form 2adx 
(x+a)(x—a) 
which we transform into 
A B 
(ear — ) de (2) 
which being reduced to a common denominator becomes 
Ax—-Aa+Bx+Ba 
(j@-a)eha) ~ 
Making this last numerator equal to that of (1) we have 
2a=Ax—Aa+Bx+Ba 


or 
(A+B)x+(B—A—2)a=o 
from which we obtain 
A+B =o 
and 
B—A—2=0 

whence 

A=} and sb=—1 
Substituting these values of A and B in (2) we have 

2aax ax ax 


x?—a®” x+a xta 


PRINCIPLES OF INTEGRATION. 269 


and by integration 


2adx ax ax 
iE ine xa 0s: («—a)—log. (x+a) 
(168) Let us next take the fraction 


in which the factors of the denominators are x and (a* — x?) 
or x(a+x)(a—x). If we make 
a +bx* A B G 
x(a+x)(a—x) et a—x abe (x) 

and reduce the second member of the equation to a com- 
mon denominator we have 

Boe ne Axe Das bx’ + Cax—Ce" 

arx— xr x(a—x)(a+<x) 


and placing the coefficients of the like powers of x in the 
numerators equal we have 
B—A—C=6 
Ba+Ca=o 
Aad’ =a3 
The last of these equations gives 
A=a 
which reduces the first to 
B—C=a-+é 
and this combined with the second gives 


rts b 
as and 6a 
2 2 


- Substituting these values of A, B and C in equation (1) we 
have 
a> +hx* aax até até 
ax—x9*— x " 2(a—x)' (OS aad 
and by integration 
i, a Fix 


g*x— x5 


atb 


4 


at+é 
dx=alog. x——— log. (a—x)— 


log. (a +.) 


270 INTEGRAL CALCULUS. 


which may be reduced to 
a log. «—(a+d) log. (a? —x?) 


Note.— The second term of the integral must be negative ; for since d(a—-~) is 


ax ax c 
—dx,we shall have d(log. GS and hence 7—x must be the differential 


a 
of —log. (a—~+). 
(169) Let us now take the fraction 
Lae 
x* +4ax—b* 
To find the factors of the denominator we must make it 
equal to zero, and solve the equation which gives 
L=—AEV 4a? +3? 
and hence the factors of the denominator will be 


LTAaTV ga?+5 


and 

XTaA— a/4q? +52 
To: simplify the expression we will represent the constant 
part of each factor by E and F and we shall have 

x* +4ax—b® =(x+E)(x+F) 
and we may make 
x Anges B —Av--Al Bs Bk 
+ 4gax—b? x+ ae +F («+E)(«+F) 
making the numerators equal we have 
Ax+AF+Bx+BE=x 


whence 
A+B=r1 
and ? 
AF+BE=o 
from which 
E F 
A= ae and B= ea 


Substituting these values es A and B we ae 


—— XIX ees A if can ae ax 
x? + sax— 2 ie c+E = EEF xtF 


PRINCIPLES OF INTEGRATION. 271 


which becomes by integrating 


E F 
E_F |S. (w+E)—p 7 log. («+F) 
or by substituting the values of E and F 
XMX atr/4 + 6? ——_—. 
cer a eT log. («ta+V 4a? +23) 
a—V 4a? + 6 
2V/ 4a? +5? 
(170) In all these cases the factors of the denominator 
are unequal. If a part or all of them are equal the rule 
will not apply. For suppose we have 
Pxt+Ox2+Rxv*?*4+Sae+T 
(@—a)(x—0)(a—c)(@—d (eZ) 
which we make 


Api Bey aC. See 
cup aa ar + ied, 


xX—C'X 
if some of these factors are equal, say a=4=c, we should 
have 


: 


log. («-+a—V 4a? +67) 


AE 


Pxt+t etc. evn dein ees oD) i 
(x—a)?(x—d)(x—c) xa x—a" x—e 
Thus in reducing the second member to a common de- 
nominator, A-+B-+C would have to be considered as a sin- 
gle constant A’, and the three constants A’, D and E would 
not be sufficient to establish the five equations of condition 
which are required in making equal the coefficients of the 
like powers of x. Inorder to avoid this difficulty we decom- 
pose the original fraction and make 
Px*+Qx3+ etc. Set Bx+ Cx? D E 
(x—a)*(x—d)(x—c) ss (x —)8 x—a! x—e 
which contains the necessary number of constants, and at 
the same time, when reduced to a common denominator, 
will produce a numerator containing «x to the fourth power; 
thus giving a sufficient number of equacdions between the 
coefficients of the like powers of x, 


272 INTEGRAL CALCULUS. 


In the meantime the expression 


A+Bx+Cx? 
\~—a)? 
may be put into the form 
A’ B’ G 


(x—a) (ea) Tea 
in which A’, B’, C’ are determinate constants. For let 
x—a=u then x=u-+a 


and 
A-+-Ba+Ce* At Ba-+-CUe* - Be s-2Ca7- ae 
(x—a)> u 
Cree ise’ 
a u® hod Py, 


and replacing the value of wz we have 
A Bas- Cae A+ Ba Cas ae Sr 
(x—a)® ——— (x—a) (x—a)? *x—a 
and since these numerators are constant we may represent 
them by A’, B’, C’, which gives 
Asp Bae Cay wa en B’ es 
(G—4)2) 7 (aa) 3b aye ee 
which is the required form. 

As this demonstration may be applied to an expression 
containing any power of x, we make the proposition a gen- 
eral one, that 

Dig = (20012 rare tes sec ca 
(x—a)™ o 

A A’ AY 
(w—a)"™ (~—ayn—t Te ayn® 


Hence to integrate the expression 


+ etc. 


__ Pxt+Qx*-betc. 
(x—a)? Ca) Ie = 
we write 
(2s ee) 
(x—a)?* (xa)? tx—a tad 


PRINCIPLES OF INTEGRATION. 273 


and reduce these fractions to a common denominator and 
find the values of A, A’, A”, D, E, in the manner already 
stated. We shall then have to find the integrals of the fol- 
lowing expressions, 
E D A’ A’ | A 

ieee te ee ee ee 

x—e x—ad x—a~? (x—a)*”~” (x—a) 
the three first we can integrate by the rule for logarithms 
and the others as follows. 

Since dx is the differential of x—a we will make x—a=z ; 


then we have 


Adx Adz A A 
ieee =| oe = JAS ae era ais aye 
and 
oe A’dz A’ A’ 
=e p= {Grey =a 
Hence 
f Pxt+Qx3 Pete. A ACO) 
(x—a)3 EEN ee ~ 2(x—a)® x—a { 
+A" log. (x—a)+D log. (x—d)+E log. (x—e) 


(171) Let us take for example 
Eas 4 


x3 —ax*—a*x—a® 

the denominator of this fracticn may be resolved into the 
factors | 

(x? —a?)(x—a)=(x—a)(x+a)(x—a) 
or 

(x—a)?(x+a) 
Making then 
a A A’ 

 (x—a)? 1)? (aa) (x—a)? oT ea beta CR) 
and reducing the second member to a common denominator, 
we have 


x? A(w+ta) +A'(e®—a*) + B(a—a)? 
(x—a)*(x+a) (wa)? (w+) 


274 INTEGRAL CALCULUS. 


Developing the numerator and making the coefficients of 
the same powers of x equal, we have 
A+B=1, A—2Ba=o, Aa—A’a?+Ba?=o (2) 
from which we obtain | 
A=ta, A’=?, B=} 
hence 
6 ch Ch adx 30x 


5 


py If all the pntae of the denominator are ana the 
expression will be of the form 
mlx 
(a—a)” 
and we may integrate this more directly. For let x—a=z, 
then de=dz and x=z-++a, hence 
pe ree eae 
( a= a) " gin 
Expanding (z+a)”~1! by the binomial theorem, we have 
flzeices oats Rela 1 ag" "az ) | 


m gin 


———4- etc. 


each of which terms may be integrated separately as in Art. 
(160) (162). 
Let us tak* for example 
a ae 

(xa)? 

Making x—a=z, then x=s+a and dx=d2, we shall have 
xa (2z-+a) eee eee 
(x—a)>— 8 age anes 2 


and 
paue 20 eo. 2a a" 
2° g 


= log.z——-— 2 —log. (SE Feo ae epee 5 


24 


PRINCIPLES OF INTEGRATION. 275 


({73) When two differential functions are equal to each 
other it does not necessarily follow that their integrals are 
equal; but if they are not equal their difference will be a 
constant quantity for all values of the variable. In other 
words, if two functions have the same rate of change they 
will either be equal to each other constantly, or else their 
difference will be constantly the same. Thus the ages of 
any two persons increase at the same rate, and they will be 
therefore of the same age, or else the difference of their 
ages will always bethe same. ‘Two persons traveling in the 
same direction, at the same rate, will either be constantly 
together, or else there will be constantly the same distance 
between them. 

Hence in integrating the members of a differential equa- 
tion, it becomes an important part of the problem to ascer- 
tain if there be any difference between the integrals, and if 
so, what it is. 

To do this we add the indeterminate constant C to one 
member of the integral equation, which shall represent this 
difference if any. This is called the indefinite integral. 

Then, since the difference is constant for all values of the 
variable, we assign to the variable in one member of the equa- 
tion some value which will correspond to. a 4zown value of 
the other entire member, and thence obtain the value of C. 
That value having been substituted in place of C in the 
integral equation, will satisfy it for every value of the vari- 
able since it does for ove value. 

To illustrate these principles, let us take the triangle ABC 
(Fig. 30). The differential of the surface of this triangle is 
axdx (Art. 62); a being the tangent of the angle CAB, and 
A the origin of coordinates. Hence the equation 

@S =axdx 
and integrating each member, and adding C to the second, 
we have 


276 INTEGRAL CALCULUS. 


ax? 


ge 


Tae (1) 


<= 


Now to determine the value of C we give to x a value cor- 
responding to a known value of S. But we know that at 
the origin in A where xo, we have also S=o, and by sub- 
stituting these values in equation (1) we have 

o=o+C hence C=o 


and 


is the definite integral. 

If now we wish to know the value of any specific part of 
the triangle, such as ADD’, we make x=x’=AD, and we 
have 

Pat Ge « ADA DD: 


2 2 2 


This is the specific integral. 


(174) We are not bound, however, to make the value of 
S commence at the origin where x=o. We may if we choose 
estimate it from any line as DD’. In this case (making 
x=AD=2’') we should have 


/ 


ro E 
ax ® . 
eines TREE rt D 
whence 
. ax ® AD a D 3 
C=— 2 Og 2 Fig. 30. 
and substituting this value in equation (1) we have 
: 2 
ied AD 
5 =a 
2 2 


This is again the definite integral. For any portion of the 
triangle estimated from DD’ we give the corresponding value 
of x, say x=AE=.", which gives 


PRINCIPLES OF INTEGRATION, 277 


Wiens 


for the value of the area DD’E’E. 


(175) There is another method of disposing of the inde- 
terminate constant, which consists in giving to the variable 
two definite values, and then subtracting one integral from 
the other. This is called integrating between limits. Thus 
ir the case last noted, if we make x successively equal to 
x’ ==AD and x” =AE, we shall have 


"9 "9 


ax 
-+G@ and 9.—=—_ -+¢ 


2 2 


aX 


Si 


and subtracting the first equation from the second we have 


ABA AD? 


ee ene ome PONT 
SS ee )=a 


the constant C having disappeared in the subtraction. 

The notation for this kind of integration consists in plac- 
ing the two values of the variable at the extremities of the 
sign of integration; thus 


on" 
AXNQN 
a! 


indicates that the integral is to be taken between the two 
values of x represented by x” and x’; the subtractive one 
being at the lower extremity of the sign; and the integral — 
.would be 

ox"? ax’? 


2 2 


When the integral is to be taken for any particular value of 
x, as x’, it would be written 


{ p= CHEK 


which indicates that the integral is to be taken where x=2’. 


278 


INTEGRAL CALCULUS. 


EXAMPLES, 
fix. 1. . Integrate 2xdx between the values of «=a and x=0. 
Ans. p*—@? 
. . b 
Ex. 2. What is the integral of tf Bx" ax, Ans, 0% ae 
a 
d= m4 2 
Ex. 3. Integrate ieee Ans. Gla—a?) 
: | 
Ex. 4. Integrate i 2(e+x)dx. Ans. 6? +2e(b—a)—a? 
a 
b 
4,5, autegrate f 3(etnx*)* 2nxdx. 
= a 
Ans. (e+nb*)§ —(e+na?)* 
0 ax e+b 
“x, 6. . Integrate i essen Ans. log. 
(176) INTEGRATION BY SERIES. 


If it be required to integrate a differential of the form 
F(x)¢@x in which F(x) can be developed into a series, the 
approximate integral may often be found by (Art. 164), and 
if the series is rapidly converging, its true value may be 
nearly reached. Let the differential be 


ax 
Stine =(1+*)-1lax 


developing by the binomial theorem we have 


(1+%?)-1=1—x? +4—<x® + etc. 


and multiplying by dx and integrating we have 


SOX, 


(eee eh 
gt eae PRE agp Tp ae 


s 
re) 
EXAMPLES. 


. e a: 
1. What is the integral of ae Ans, 
I+<x 


PRINCIPLES OF INTEGRATION. 2709 


ax 
Ex. 2. What is the integral of 77? _ Ans. 
rn ax 
Ex. 3. What is the integral of e=eae Ans. 
ve Ae Woot re theuntepralot ei ? Ans. 


Vi —x* 


(177) INTEGRATION OF DIFFERENTIALS OF CIRCULAR ARCS. 


We have seen (Art. 47) that if ~ designate the sine of an 
arc, then the differential of an arc will be 
Lu 
Vi—u? 
hence the integral of the function of the form 
ax 
V1—x? 


will be an arc of which x is the sine. 


(178) If the expression is of the form 
ax 
V a — x? 
we may make x=av then 


ax*=a*v® and a®?—x* =a?—a?y* =a?(1—v?) 


and 
adx=adv 
whence 
ax eet a eae 0 
4a a8 © a1) a 
and. 


fi ax = av 
V a®—x? Vi—v? 


which is an arc of which 0m Ie the sine. 


280 INTEGRAL CALCULUS. 


(179) If x represent the cosine of an arc, then the dif- ° 
ferential of the arc (Art. 47) will be 
ax 
Vi—x’? 
hence the integral of the form 
aX 
V1—x? 
will be an arc of which ~« is the cosine. 
If the expression is of the form 
aX 
V a? —x? 
it may be integrated as in (Art. 178) 
(180) If » represent the tangent of an arc then (Art. 47) 
the differential of the arc will be 
Ly 
Lope" 
hence the integral of a function of the form 
ax 
I+x%* 
will be an arc of which « is the tangent. 
(181) If the expression is of the form 
ax 
a? +x? 
we may make x=azv, whence 
dx=adv and a®+x*2=a* +a*v* =a?(1+0*) 


whence 
f{ ax f adv af wv 
a®*+ 42° J a®*(1+0*) ad 140? 


Se aes Li : aie, 
which is equal to 7 into an are of which Tease the tangent. 


(182) If we represent the versed sine of an arc by z we 
have (Art. 47) for the differential of the arc 
Us 


—————— et 


/23—3? 


PRINCIPLES OF INTEGRATION. 281 


‘hence the integral of a function of the form 


¥ 


ax 
V 2x—x? 
will be an arc of which x is the versed sine. 
(183) If the expression be of the form 
ax 
‘/ 2ax —x? 
we may assume x=av, whence 
ax=adv and 2zax—x* becomes 2a?v—a*v2 
or : 
a*(2v—v?) 
whence 


ax 
(lel jbl 


which is an arc of which Us is the versed sine. 


SECTION II. 


INTEGRATION OF BINOMIAL DIFFERENTIALS, 


(184) The general expression for a binomial differential 
may be reduced to the form 


D 
a™l(qtbxr) id (1) 
in which # and z are whole numbers, zis positive and x 
enters but one term of the binomial. 

For if m and z are fractional we may substitute another 
variable with an exponent equal to the least common mul- 
tiple of the denominators of the given exponents, which will 
then be reducible to whole numbers. 

If, for example, we have 

1 me fe 
x3 (a+dx*) dx 
we make x=2°, then dx=06z'dz, and we have by substitu- 
tion 


ose 
627 (a+b2°) Vaz 
: : I 
If 21s negative we can make x=, and the expression 


would become 
Dp 


£ 
am Mgt hxe-n) I =2-M-V(g+b2") Tax 
in which the exponent of z within the parenthesis is pos- 
itive. 


282 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 283 


If the expression is of the form 
2 
ae yt iden a ee 
we may divide the terms within the parenthesis by x”, and 
multiply the parenthesis by it, thus 


# 
Ml xt (aban) | Vax 
or 


m—1 - 2 
— + —_ = _— 
Bs 9 (atdx"-") Vax 


thus we may secure the three stated conditions. 


(185) If : is a whole number and positive, the binom- 


ial may be expanded into a finite number of terms and 
integrated by Art. (163). If it is entire and negative the 
function becomes a rational fraction. 


EXAMPLES. 


£x.1. Integrate the expression 
x*(atbx?)* dx 
Expanding the binomial we have 
a*x*dxt+2abx'dx+b* x8dx 
and integrating each term separately we obtain for the in- 
tegral of the binomial differential 
Cad MDNR OP ee 


Sx? (atbx?)* dx ied + 


8: 9 
Ex. 2. Integrate the expression 
x3 (a+tbx* )3ax 
aia emer pee yoy OKO 
Ans. Soe omer a4 oars 
6 8 IO 


x. 3. Integrate the expression 
x4 (atbx3)3 dx 
ax®  3a%bx8  zqb®x11  Y3yt4 
8 II 14 


Ans. 


284 INTEGRAL CALCULUS. 


Lx. 4. Integrate the expression 
inate 
ae ghee zabtxt4 fb~18 
14 is 


Ans. aes aig 


(186) Zvery binomial [baa may be integrated when the 
exponent of the variable without the parenthesis, increased by one, 
zs exactly divisible by the exponent of the variable within. 

To effect this we substitute for the binomial within the 
parenthesis, a new variable having an exponent equal to the 
denominator of the parenthesis; thus in the expression 

Y 


xl q+hx”) Tax (1) 
we make 
atbx™=29 (2) 
then 
PD 
(atbx") 1 =P (3) 
From equation (2) we have 
1 
veg hr) Nt 
a=( 5) 


and raising both members of the equation to the mth power 


we have 
m 


xm=(™ —“\" 


Differentiating and dividing by m we have 


mM” 
hows 


q (24-4 Ey 
m1 A—*ds 
a Lae ja mms ( (4) 


and multiplying together equations (3) and (4) we have 


z gee 

' f. g £2 —@ ; 
vires —_4__ —1 a 
xn (a+ bx”) Tdx=— aera ( I ) “ 


mM. . ; os : 
If now 7 1S an entire positive number, this expression may 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 285 


si—e 

b 
limited number of terms, and each term can be integrated 
separately. 


be integrated by raising to a power consisting of a 


2 . 
If be negative we may by formula D (Art. 214) increase 


the exponent until it become positive. 


EXAMPLES. 


(187) Integrate the expression 


3. 
x3 (atbx*)* dx 
Assume 
atbn® =32* 
then 
3 
(at bx”)? =28 (x) 
2° —@ 
ax = ai (2) 
and) 
£a2 
xan =F | (3) 
Multiplying (1), (2), (3), together we have 
3 2" —@ 
«3(a+bx)*dx =24 ads 


of which the integral is 


B 
stags (at bu?) a(atbx*)? 


Borat D? ean c 50° 
({88) Integrate the expression 
5 (a-+bx*)*dee 
Make 
atdx* =" 
then : 


(a+bx2)® =z (2) 


286 INTEGRAL CALCULUS. 


2°—a 
oes 
Kiet tay ee, (2) 
and 
2d 
xdx = (3) 


Squaring (2) and multiplying by (1) and (3) we have 
- Z\ ®2°d2 28 dz—2az4*d2-+a" 2" dz 
b Ee ee De 


st g 
ee ee) Cade ( 


of which the integral is 
2  2az% a*z 
iB 58 Tobe 
and restoring the value of z we have 


ue 5. 3B 
(at dx?)® 2a(atdx*)? a?(atdx?)? 


i 
Sx (atbx*)*dx= 6 aE 303 
(189) Integrate the expression 
es 
x®(atbx") sax 
Make 
atbx* =z3 
then 
2 
(a-+0a*) =e! (x) 
also 
: Saat 
Cy, (2) 
and 
2 
ae (3) 
Multiplying the square of (2) by (1) and (3) we have 
327728 — ay®., 
x(a-+dx*) dx =" ( F ) ade 
324 
=F pl2' — 288 ata? az 


391%7e 25%agem eo eds 
2032 age ay 26° 


—_— 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 287 


which being integrated is 

Sous esas aed 

2263 863 ' 1083 
Substituting the value of z we have 

8 
one, . 3latdx*)4 3a(a+dx*)® 3a* (a+bx?) 
FONT IESE remarry | mn eT Sn 
(190) Integrate the expression 


ae 
x3 (a+x*) *ax 


5 
3 


Make 
atx =z 

then 
x*=2%—a¢ (1) 
(atx? ee (2) 
xan = s2dz (3) 


Multiplying together (1), (2), (3), we have 
=o 
xsdx(atnx?) *=(2?—2)2-ledz= (2? —a) dz 
and integrating we have | 


3 
23 eye 
te ala +228 


S 
(191) Integrate the expression 
x(a? +x7)-1ax 


Make 
a* +x*=2 
then 
x* =s—<a? (x) 
2xdx=az (2) 
and : 
(a? +x?)-1=2-1 (3) 


Multiplying together (2), (3) and the square of (1) we have 
72 \2o—1 
x(a? Beet eae : : gees 


which being expanded becomes 


288 INTEGRAL CALCULUS. 


2° —va*ztat sdz 2a*dz atds 
ee ee 
2 2 2 25 


Integrating and substituting the value of z we have 


Week ENO 

Sx® (a? $a?) te EY 

({92) Another condition under which a binomial differ- 
ential may be integrated is as follows: 
wy 


Put the expression «”-1(a+dx”)9d¢x into the following 
form 


4 
—a*(a? +x") +log. (a? +x?) 


3 P 
x1] (<4 0)xr] Cax 


or 


np np i 


= - m + 1 - 

xl @ (+5) 1an=a0 Y ~ (ax-"+b) dx 

xn 

By (Art. 186) this expression integrable when 
np 
m+—— 
eet sis & aiii 
1 ht -@ 

is a whole number, hence 
A binomial may be integrated when the exponent of the vari- 
able without the parenthesis, incréased by one, divided by the 


exponent of the variable within the parenthesis, and added to the 
exponent of the parenthesis 1s awhole number. 


EXAMPLES. 


(193) Integrate the expression 


a5 
a Tapes) aor 
Make 


then 


INTEGRATION OF BINOMIAL DIFFERENTIALS, 


also 
I 
x pe 
whence 
Uae 
ce ~ a(v®—1)? 
and 


1=x4(v?—1)? 
Multiplying together (1), (2), (3), we have 


= adv 
ai+x*) x= 


and by integration 


(194) Integrate the expression 


ax( 2 2) ax 
6 Phd SE 
ane V a? +x? 
Make 
V=XTV a? +x? 
then 
toe er iere 
fo=te + — ite Sea Mogae aS 
V at + x? V a2? +x? 
hence 
Go ax 
0 MV Gtx? 


Representing the integral sought by X, we have 


289 


(2) 
(3) 


= ax Ly a 
X,=f hee ap log. v=log. («+V a? qe), 


Uv 
(195) Integrate the expression 
ne 
V/ a? +x? 
Representing the integral by X, we have 
y= Lee 
OOM hae 


290 INTEGRAL CALCULUS. 


Make 
v—(a? x? +o)? 
then 
_@ SAUL ee een ao Oe oe 2x* dx 
(a? x? tat)? Sverre AYR oT 
hence 


du=a2dX,+2dX, 
where X, has the same value as in (Art. 194). From this 
we have 


dv a*dX 
GPS aR 
or 
vy ax 
SS eee 2 


Replacing the value of X, and v we have 
1 2 
X, =—(B = x®)®_ log, (xn+V a? +x?) 
(196) Integrate the expression 


il: 
xrtla— xe) x 


Make 
v°x® =1— x? 
then 
x-?=y?+7 and x-4=(v?+1)? (x) 
also 


x=(v? + yt whence dx=—(v?+ 1) 2 ado (2) 


and 
1 
a1 or {v? +1)? 
(12?) eee ee ae (3) 
Multiplying together (r), (2), (2 we have 


x-4(9—x?)7 Fa —(v? +1)? (vw? + je = a vdu=—(v? + 1)dv 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 2g! 


Integrating we have 


Jona) tes p= 
({97) If a binomial cannot be integrated by any of these 
methods, there are others to which we may resort. These 
consist in making such a transformation of the expression 
that the exponent of the variable without the parenthesis 
or that of the parenthesis itself may be reduced so as to 
bring the differential into one of the integrable forms. This 
is done by separating the differential into parts, one of 
which shall be an integral quantity, and the other the form 
to which we desire to reduce the expression. This is called 


INTEGRATION BY PARTS. 


To effect this we resort to the principle on which the pro- 
duct of two variables is differentiated. We have seen (Art. 
11) that | 

auv)=udv+ovdu 
hence 
uv=fudo+fodv 
or 
Sudv=uv—vdu (1) 

If now we can so transform the binomial differential, that 
while it is represented by the first member of the equation 
(1), it may also be represented in its transformed state by 
the second member, we see that the integral may be made 
to depend on that part represented by vdu, and that may be 
made to assume in certain cases the form of an integrable 
differential. | 

The two general methods of doing this are, either to make 
the part represented by to contain the variable without the 
parenthesis with an exponent diminished by that of the 
variable within the parenthesis; or else to contain the par- 


292 INTEGRAL CALCULUS. 


enthesis itself with an exponent diminished by one. In all 
other respects this part is to be identical with the given dif- 
ferential binomial. 

The following is the first of these methods. 

For convenience we represent the exponent of the par- 
enthesis by /, which is supposed to represent a fraction; and 
substitute # for #m—1; and we have the general form 

x” (atbx”) Pde 
in which # and z are whole numbers. 
Make 
v=(atdbx" )8 
in which s may have any required value. Differentiating 
we have 
dv=bnsx"-\(a+bxn")s§ldx 
If we now assume 
x™ (a+bx")P dx=udo 
we have 
x (atdx”)P dx 
“—~bnsx*® a + bx”) lide 
or 
am—n+l(gthxn)p—s+t 
see ons 


which being differentiated gives 


ioe, xm—n bxn \p-s+1 
Spall n+r) (a+ a 


bns 
4 { Jost er eae 
but 
(atdx”)P-8+1a=(atdxe")P-%(atdx”) 
=a(a+dbx #\P—#1- fog h (atbx”)P-s 
hence 


Am—ntr)x"—" (mtr +nup—ans jx” 


as 7S 


|(c + bx") P-8 dx 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 293 


If now we take the value of s such that 
m+1+np—ns=o 

we have 

_ mrt 


nN 


+p 

and 

alm—n+1)x"-" (atbx")P-s 
| (np+tm+1) 
Substituting these values of w, v, de and dv in equation (1) 
we have 


lu be 


ForMULA A 


Sa™atbx*)Pdx= 
ime a Ce oa aaa Pon Pde 
{ b(np+m+z) 


in which we find the integral of the given differential to 
depend on the integral of a similar differential in which the 
exponent of the variable without the parenthesis is dimin- 
ished by that of the variable within it. 
In like manner we should find 
Sx?“ atbx®™)? dx 
to depend on 
Sx athe)? ax 
and we may thus continue to diminish the exponent of the 
variable without the parenthesis as long as it is greater than 
that of the variable within it. 
(198) There is frequent occasion to integrate binomials 
of the form , 
Sa eg 
Representing its integral by Xm we pee 


Xn= | 75s 


— 


294 INTEGRAL CALCULUS. 


and substituting in the formula A (Art. 197) 
—1 for d 
2 for 2 
a” fora 
—4 for p 
we have 


FORMULA @. 


ade el tt ae en oe mee 
Var—x® 8m Var—x®  m 
(199) Integrate the expression 
adx 


— at — x? 
We have found (Art. 47) that the differential of the arc of 
a circle is equal to 


V a®—x? 


Xm= 


aX, 


Rd sin. 
a/R? —sin.? 
hence we have 
X, =arce of a circle of which a is radius and ~ is the sine. 


(200) Integrate the expression 
ij. wae 
dF VW Gk 3? 


a” —x* 
Substitute in formula a, 2 for # and we have 


2 
a 


BO ee 
Sal x2 


v 2 at x* 
which is equal to 


9 
"od A ee 
x ——V a? — x? 
2 


where X, has the same value as in (Art. 199). 
Similarly by substituting different values for m in formula 
a we have 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 295 


=f eran ich a hos a 
RCN gi ie mre ar | 4 
wel ae a* ol, We eer Te 
sgoeetl Sen ae =e By meen Cet at 
iad at We 
Xe a 
in which the values of x. X,, X4, Xg, Xg remain the same 
throughout. ‘Thus formula @ reduces the integral of a dif- 
ferential of the form 


xt ~—--- 
= Pa vaue — x? 


Ba Gs 
V a®—x? 
to that of one depending on the integrals of differentials of 
the forms. 
a2 Tec rofl see aa xb 
until, if # is an even number, we shall after { operations 
find the integral of the given differential to depend on that 
of a differential of the form 
ax 


V a? — x? 
which is the differential of the arc of a circle of which re 


is the sine (Art. 177). 
(201) By a similar substitution in formula Awe may find 


FORMULA 06 


al eames ae cA errr 5 ee POSE VA ade Mamta Ge 
Va®+x%  ™ V a? + x? 


If then we have the expression 


x*dx 
4 =/T= 7 x? 


296 INTEGRAL CALCULUS. 


we would make m my in formula 4 which would then become 


= 3a" mepd (he 

OG = Vete— 7d ae ae 

The integral of 
Lae 
V atx? 
we have found (Art. 195) to Ps 
al 

(e+ oF — “tog («-+/a? +2") 

hence 
4x x? ax 
aaa + x2 = eee’ a aaa wee 8 V/ az +2 +x2 ae log. (a +V/ a2 +x? +22 ) 


(202) The: expression 


may be integrated by first reducing it to the given form (Art. 
184) and making the proper substitutions in formula A (Art. 
197). 

It may however be integrated by an independent process 
as follows : 

Make 


ses ane ct 
and we have by differentiating 
al 2m— 1).x*M—2L4 — iit lif 


which becomes by dividing the terms by 2-1 


ts a(2m—1)x"™l1gx mx Mio 
oP aN a ees 

(2zax—x*)? (2ax —x*)* 
Now this last term is equal to 


hence 


a(2m—1)a”"—-lax 
do aaa eh: 2 
(2ax—x*)* 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 297 


or by transposition 
a2m—1)x"-ldx dy 
ip sea 3 


max —x? 2 is 


and by integrating and substituting the value of v we have 


FORMULA ¢ 


CaaS _aam— Hiphee ate ee 
—— if ———— V 2ax— x? 
en gies HM 


Mg V2ax—x2 m 
an expression which depends on the integral of 
hele Ee 
V 2ax— x? 
in whicn the exponent of the variable without the parenthe- 
sis 1s diminished by one. 
(203) If we take the expression 


Se adx 

Oe Serra x? 

we see (Art. 47) that it is the differential of an arc whose 
radius is a and whose versed sine is x ; or, which is the same 


. ° . Xx . 
thing, an arc whose versed sine is me and radius 1; hence 


adx j ao 
= =ver. sin.—1!— 
ilar me a 


2ax— x* 


(204) If we take the expression 
oa NIK 
Lene 


2ax— x* 


and make min formula ¢ equal to 1, we shall have 
X,=Xo—V 2ax— x? 
in which X, has the same value as in (Art. 203). 
Similarly 


6; fe en 
X= fe x, FV tan 


and 


298 INTEGRAL CALCULUS. 


38 2 
a eee se ae V 2ax— x? 
Mace x? 3 3 
where X14, X, Xs, have the same value throughout. 
Thus formula ¢ reduces the binomial differential. 
xMdx 


V 2ax— x? 
to depend successively on the integrals of 
xl gx age: Colm ff 


V 2ax— x? V2ax— x?’ V 2ax— x? 
and finally on 


ee 
V 2ax— x” 
which represents the differential of an arc whose versed sine 


x 
1s 7 as we have seen. 


(205) To find the integral of 


Bed 45 3 
V/ 2AX— X 
we substitute in formula ¢, $ for # which gives 
4 2 
peas _ 4a nek 4 Ke Df Pps peal | 
poe = a V 2ax— x? 
V2ax— x2 V 2ax— x? 3 


Dividing the terms a the first fraction in the any mem- 
ber of the equation by x we have 


1 
x? dx ax 
V 2ax— x2 V2a—-x 
of which the integral is 


—2V 2a—x 


hence 


x°*dx 8a 2x 
ee ca 
5 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 299 


(206) The method of diminishing the exponent of the 
variable without the parenthesis by means of formula A, 
will of course only apply when mm is positive. But we may 
obtain from this another formula which will diminish the 
exponent when it is negative. To do this we multiply the 
formula A by the denominator and we have 

b (ptm+t1)/x™atdx”)P am 
Uo xt hat bx0")Pt1—a(m—n+1) fx" —"(a+bxr)P de 
or 
late! (a +bx" )P “x= 
xm—n+1(q+dx”)\P+1—db(nptmtr) /x™atbxe”)P dx 
a(m—n-+1) 


Making m—n=—m we have 


FoRMULA B 


ax 1(qg+hx")P+1—b(nptm+ntr) /x-™*(atbn”)dx 
a(—m+1) 

If ¢z denote the greatest multiple of z contained in m we 

shall have after 7+1 reductions the integral of 
x—™(atbx”)P dx 
to depend on that of - 
xa mt (E+ In( a +x nv \? ax 

and if —m+(¢+1)z=nx—1 we shall have (Art. 165) 
(a-+4x% pti 

nv( p+1) 


| [x-™atbdx”)Pdx= 


[x®-"(atbx" )? adx= 


but in this case 
—m+I 
2 
a whole number; and hence the original expression may be 
integrated as in (Art. 186). 
(207) To find the integral of 


300 INTEGRAL CALCULUS. 


ax 


aad (i +238 


Substitute in formula B 


1 
=xa-*(1+43) 3ax 


2 for m 
1 fora 
1 for 6 
3 for z 
—+ for p 
and we have 
ee: 2 afl he 
Ja Mite) Sde=—x Na +a*)* ali tx?) ae 
since a(—m+1)=—1 and 4(mp—m+1)=1. 
(208) To find the integral of 
aX ee) 2\—3 
Ee Gao (2—x*) *adx 
x? (2—x?)* 
Substitute in formula B 


2 for m 
2 fora 
—1 ford 
2 for 2 
—+ for p 
which gives 


fu? (2—22) Yae= = 9-1(g—<92) oe af(a—x*)\ ae 
since a(—m-+1)=—2 and d(mp—m+n+1)=2. 

(209) Besides the method of reducing the exponent of 
the variable without the parenthesis, we may make the 
integral to depend on that of another expression of the same 
form in which the exponent of the parenthesis itself is re- 
duced by one. ‘This is the second general method referred 
to in (Art. 197). 

Let us make v=x* where s is an exponent to which we 
may assign any required value. From this we obtain 

dv=sx8—-ldx (1) 


INTEGRATION OF BINOMIAL DIFFERENTIALS. 301 


If now we assume 
udv=x™atbdx”)P ax (2) 
we shall have by dividing equation (2) by equation (1) 


xN—s+ 1 
v5 omme rT, +x)? 


and 
— b 
du Seas (atdx” )P dx 4-2 am—8t1( qt hyn\P-lyr—liag 
but 
(a+dx")P =(a+bx")(atdx™)e-1 
hence 
a _am—s +1)+d(m—sti1+up)x” 5-8 at Bao Pld 


i$ 
Let the value of s be taken such that 

m—s+i+nup=o 
or 

s=m+i+nup 
and we shall have 
—anpx™—*(atbx”) Plex 
up+m+ti 

Substituting these values of w, v, du, dv in formula (1), (Art. 
197), we have 


2 


FormMuLA 6 


aM gtbhx”)P +anplx™atbx”)P-ldx 
up+tm+i 

in which the integral of the expression is made to depend 
on that of one of the same form in which the exponent of 
the parenthesis is one less than that given. | 

By a similar process this last may be made to depend on 
the integral of one whose exponent of the parenthesis is 
again one less; and so on until the exponent of the paren- 
thesis shall have become less than one. 


Sam a+ bx”) P dx = 


302 INTEGRAL CALCULUS. 


(210) To integrate the expression 
AXN g2 + x2 
substitute in formula © 
o for m 
a” for @ 
1 for d 


and we obtain 
xa/G2ge a2 doc 
but by (Art. 194) we have found 
ax 
| Teepe Vara) 


hence 
‘ee en) 42 2 2 ae 
(Ax/ Q® + x3? ee ee +—log. (e+ ar x?) 
in like manner we find ; 
——  *Vx?—@? ag ae 
Sdx/ x2 — g? = ae aes (e+ x? — a”) 


(211) If the exponent of the parenthesis is negative, this 
formula will, of course, not answer, but we can easily deduce 
from it one that will effect the object. For this purpose we 
clear it from fractions, transfer the integral term, and divide 
by the coefficient of the last term in formula € and we have 


Formuta D 
ie Matbx n\ D-lile 
=a Wat ben)? + (up-tmtr) /am(atbar )P de 


anp 
(212) To find the integral of 


Eg 
(2—x*) "dx 


INTEGRATION OF BINOMIAL DIFFERENTIALS, 303 


substitute in formula D 
o for m 
2 fora 
—1 for d 
2 for 2 
—3 for p—1 
and we have 


2 =} 
[ (2—x? 2 dx=—(2—x2) P= 
2 


(213) To find the integral of 
xAX 
(a +23)3 
we substitute in the formula 
1 for m 
1 fora 
1 for d 
3 for z 
—+t for p—1 


x 


2 2— x 


=x(1 4 8) Bx 


and obtain 


Sx(1 4x3) tax — 


2 


x 


(a Bays eat fecl ¢ 4 2) Ste 


2 


in which «(1-+.v?)%¢x may be developed into a series, and 
each term integrated separately. 


SECTION III. 


(214) Application of the Integral Calculus to the Measure- 
ment of Geometrical Magnitudes. 

We have seen (Art. 173) that when two differentials are 
equal, their integrals will also be equal or else have a con- 
stant difference. It is upon this principle that the method 
of measuring geometrical magnitudes by means of the cal- 
culus is founded. We obtain the expression for the rate of 
change in the magnitude, in a function of one variable and 
its differential. It will follow that the magnitude itself is 
equal to the integral of the function, or else the difference 
between them will be constant for all values of the variable. 

Thus let M represent any magnitude, and let F(x)dx rep- 
resent its raze of increase while being generated by its ele- 
ment—that is, its differential: F(x) being the differential 
coefficient, and a function of x. 

Then we have 

adM=F(x)dx 
which is an equation between two differentials, hence the 
integrals are equal or else differ by a constant quantity. If 
we represent the integral of F(x)dx by X we shall have 
M=X-+C 
where C represents the constant difference between the 
quantities whose rates of change are equal. 

The method of disposing of the term C is shown in (Art. 

173), and the result will be an expression for the value of 
304 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 305 


M in terms of one variable. Then assigning to this variable 
any specific value, we obtain the value of M from the be- 
ginning up to that value of the variable; or, by giving to 
the variable two successive values, the difference of the two 
resulting expressions will give the value of that portion of 
M lying between the two values of the variable. 


RECTIFICATION OF CURVES. 


(215) To rectify a curve is to find what would be its 
length if it were developed into a straight line; in other 
words, to find the measure of its length. When its differ- 
ential can be obtained in an integrable form it is said to be 
rectifiable. 

The general expression for the differential of any plane 
curve whose equation is referred to rectangular axes is 


(Art. 34) 
Mu=N/ dx? +dy* 

and hence 

u =/V ax? -dy* +-C 
is the general expression for an indefinite portion of any 
such curve. In order to obtain the integral of this expres- 
sion, we must know the relation between x and y which we 
obtain from the equation of the curve; and by means of it 
eliminate one of the variables and its differential from the’ 
formula; thus producing a differential function involving 
but one variable and its differential, whose integral, when it 
can be obtained, will be the length of an indefinite portion 
of the curve. 


(216) To find the length of a Circular Arc. 


We have in (Art. 47) several expressions for the differential 
of an arc of a circle, in terms of its trigonometrical func- 
tions, which already contain but one variable in each case. 


306 ° INTEGRAL CALCULUS. 


If we select that in which the tangent is the variable, we 
will represent it by ¢ and the formula becomes 


at T 


lu = ari jemi Tape 


Developing the fraction we have 
I 
———__ —;— 72 AM dy 9 aus 

see ge en 


hence 


i w= LP a 7°d¢t+ete. 


A 
and integrating each term separately we have 
ye ge A ie 


M—=4u=t—— — — CLG: 
Metra CaN yn ty 
or 
2 74 76 78 
a fiero dar ie 4" ete J+C 
But we have found ae st x =o, ac if we assume 
Ua 20m 
we shall have 
Var 
and substituting this value for ¢ we have 
I I I 
ga Now i(1———+- —— Ge ete 


9 FOU Os Od eeie bec meee aia 
which being reduced is equal to 0.523598 nearly, for the 


length of an are of 30°; and multiplying by 6 we have the 
arc of a semi-circle equal to 3.141588 when radius is 1. 
Hence this is the ratio between the diameter and the entire 
circumference. 


(217) Zo find the length of an Arc of a Parabola. 


We have found (Art. 57) that the differential of an arc of 
a parabola is 


ium PHP 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 307 


The integral of this (Art. 210) is 


0 be Sr ere k Fae ats 
eh Desig oe Oe aay Deby") EG 
If we estimate from the vertex of the curve where ~=o and 
y=o we shall have 


=F log. p+C 


hence 


10g. p 


Substituting this value of C we have for the definite integral 


va ZV PFS +4 log. (WV P+ PF log, 2 


a 


or 


eee yt pep y* 
ea one? +7; log. > 


(218) Zo find the length of an Arc of an Ellipse. 


We have found (Art. 57) that the differential of an arc of 
an ellipse is 
pee TN, A4—(A?—B?)x? 
au = Tai cae 
If we take ¢ to represent the distance from the center to the 


focus of the ellipse we have 
B2=A2—-2 


hence 
I Ee CAS 
Guia Ae, gare Ses g s 
A data bet 
If now we represent the eccentricity of the ellipse by e we 
have c=Ae, and hence 


Bao A4—A2e2 x? 
du= aN] ee 8 spam 


308 INTEGRAL CALCULUS. 


. 


or, dividing by A* under the radical and multiplying by A® 
without it we have 


giant 
i 
LL A? 
du BN eee 
A*—x 


Bee A 
Developing (1 — A2 ) by the binomial theorem we have 


vo 


Bie Pre Bese Bee xe 
(e= rep, “i gAP an a A Ned oy en ne 
hence 
(= Adx e? Max e# xed | 
| VA? — x? 2A VAP? 2.4A3" A232 
gow xO6ax 
rey ms Sree 
Making 
Awe 5 oC ane Rt hoe 
Aiea Rae V RE GE LAY etc 
we have 
22 es 206 
Jdu=Xy— aXe—F gas iw GAB x6 —ete. (1) 


Now by (Art. 199) X)= the arc of a circle of which A is 
the radius and ~ the sine, and (Art. 200) 


6 Ae eR ee 2 eS 
Xo Se Rey Be 


also 


and 


If we make x=o and estimate from the extremity of the 
conjugate axis, we have x=o and C=o. If we make x=A 
we shall have w= a quadrant, and since 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 309 


we have 
A 3A? 3A Se 5k 
X,=FXo. Seal Se, pet a ae 6x0 
and substituting these values in equation (1) we have 
ro SU Dip hss hat ARE Aine es TS Age 
yh Saath AAs | ate 4 Oe 6 ete.) 


for one-fourth of the circumference of an ellipse; X, being 
one-fourth of the circumference of a circle of which the 
diameter is equal to the major axis of the ellipse. 

Hence the whole circumference is equal to 


A ae Weave sie 
eee ie Seen ates 8 SNE AN 
Pet Dees AN Ales Rel aA SAO AO 
It will be seen that as the eccentricity diminishes the cir- 
cumference of the ellipse approaches the value. of 2z 


which it reaches when eo, and the curve becomes a circle. 


(219) Zo find the length of the Arc of a Cycloid. 


We have found (Art. 129) that the differential equation of 
a cycloid is 


Vy 
= 
V ary—y? 
By substituting this value of @x in @x in the f formula we have 
du=V dx® +dy? = LONe 
2ry—y* 


or 


225: a 
If we estimate the arc from D (Fig. 55) where y=2r.we 
shall have w=o and C=o, and hence making y=FG 

“=D! F =—2V ar (27—y) (1) 
We see from the figure that 
D'E’=27 and D’'H’==27—y 
hence 


310 INTEGRAL CALCULUS. 


V 2r(2r—y)=V D'E’ . D'H’ =D'F’ 
so that the arc of a cycloid is equal to twice the corrv-_- 
ponding chord of the generating circle. 

If we take the arc D’A, the corresponding chord of the 
generating circle becomes the diameter D’E’, and half the 
arc of the cycloid is equal to twice the diameter of the gen- 
erating circle, or the entire arc is equal to four times that 
diameter. Thus, if we make y=o we have 

u=4r 
or 
D FA=2D' Gand ADB=s bere 


(220) Zo find the length of the Arc of a Logarithmic Spiral. 


We have found (Art. 77) that the differential of an arc of 
a polar curve is 
du=V ae bar® 
and the equation of the logarithmic spiral is 
v-=Log. rv 
Hence 


Mar M2ar? 
2 


a — and av? = 


Substituting this value of dv? we have 
du=NV Mar*® + dr? =drV M*? +1 
In the Naperian system M=1 and 
au=arn/ 2 
hence 
“t—ra/ BC 
If we estimate the arc from the pole where =o we shall 
have C=o and 
u=rr/ 2 
That is, the length of an arc of a Naperian logarithmic 
spiral estimated from the pole is equal to the diagonal of a 
square of which the radius vector is the side. 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 3 Us 


(221) Zo find the length of an Arc of the Spiral of Archimedes. 


The equation of the spiral (Art. 84) is 
r=av 


: : I : : 
in which a= a and v= the arc of the measuring circle whose 
4b 


radius is the value of ~ after one revolution. Hence 

2 Rn Pe AY ED 
the integral of which may be found in (Art. 210). Substi- 
tuting 1 for a and v for x, thus 


Bas 


ua—a 


+tlog. (v+V 1+7'?) ) +C 
Estimating the arc from the pole where v=o we shall have 


C—oaand 
. f= -ai eee 
eaves ee *+log.(v+V 1-+0 )| 


(222) Zo find the length of an Arc of a Hyperbolic Spiral. 


The equation of this spiral is (Art. 86) 
ab 
rv=ab or See 
Differentiating we have 
abv 


2 


adr = — 
v 


whence 


ab*dv® ab? , g ae OP Ot aN fet 
du=\f— + ae mNee 


u=abfv-* avn y? +1 
Integrating by formula B (Art. 206); making in the formula 


and 


m2 
Goat 
at 


312 INTEGRAL CALCULUS. 


we have 


Jo? aov v® +1 =—0-'(1 +0°)8-+ 2fao(t +v?) 
or (Art. 221) 


=—y-1(1 Busy paises +4 log. (otvV I $2") | 
hence 


u=abl—v-1(1 +y2)t +o itv? + log. (v+tV1+v77)|+C 

Estimating the arc from the point where v=o we have 
u=ab(t Fao : 

which is as it should be, since from the equation of the 
curve, when v=o the radius vector is infinite. As v is infi- 
nite when =o we shall have w=o at the same time. 
Hence the curve is unlimited in but one direction. We may, 
however, find the length of any intermediate portion by sub- 
stituting the two corresponding values of v in the integral 
function and taking the difference of the results. 


(223) Quadrature of Curves. 


The quadrature of a curve is the process of finding the 
measure of a plain surface bounded wholly, or in part, by a 
curve. 

To find the area of such a surface we must find its dif- 
ferential in a function of one variable, which being integrated 
will give an expression for an zzdefinite portion of the area, 
from which any sfecéfic portion may be obtained by assign- 
ing corresponding values to the variable. 


(224) To find the area of a Semt-Parabola. 


We have (Art. 65) for the differential of the surface of a 
parabola + 
AS =ypdx=/ 2px" ax 
of which the integral is 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 313 


SHB yx" = hay ype=Fay+C 
But when x=o we have S=o, and hence Co; so that 
the surface of a parabola bounded by the curve, the axis 
and an ordinate is equal to two-thirds of the rectangle 
described on the ordinate and corresponding abscissa. 


(225) To find the area of any Parabola. 


The general equation of the parabola is 


y” =ax 
from which we obtain 

n—1/7, 
(ee 2 

a 
hence ; 

ny Cay ee? 
S El “(n+1)a_ = vtec 


If we estimate the curve from the origin where S=o we 
have x=o, and hence C=o, and 


n 
Ser awe ye 
That is, the area of that portion of any parabola, bounded 
by the curve, the axis and the ordinate, is equal to the rec- 


tangle described upon the ordinate and corresponding 


: ee 5 7 ; 
abscissa, multiplied by the ratio es If 2==2, as in the 


common parabola, we have 


S=xy 

If ~=%, as in the cubic parabola, we have 
S= xy 

If z=1, the figure becomes a triangle and we have 
S=tey 


or half the base into the height. 


(226) To find the area of a Circle. 
We have (Art. 63) for the circle 
ydx=AxXV/ R2—x? 


15 


314 INTEGRAL CALCULUS. 


Making R=1 we have 


<< t 
ad S=ydx=dxV 1—x* =dx(1—x*)? 
Developing the binomial and multiplying each term by av 
we have 


BERS ONAAS OO AR ee 


53 =ax-— goa gran he Reels 
from which we obtain by integrating each term separately 
Ere Ore tata 5x9 
Di iatp ene ouNGR Tena oe 


Estimating the area from the center where x=o we have 
S=o, and, therefore, C=o, so that the series expresses the 
area of any segment between the ordinate at the center 
where «=o and the ordinate corresponding to any other 
value of x. Hence if we make x=1 we have the area of a 
quadrant equal to 
eet cere ee it tires weg eee 
which by taking enough terms may be reduced to 
-78539 

Hence the entire area of the circle will be equal to 

| 3.14156 
equal to z where radius is 1. _ 

(227) We may also find the area of a circle by consider- 
ing it as being described by the revolution of the radius 
about the center. In this case the radius of the circle be- 
comes the radius vector and we have (Art. 82) 


where v represents the arc of the measuring circle and R its 
radius. Integrating the terms of this equation we have, 
since 7 1s constant, 


TSRaR 
Estimating the area from the beginning where S=o we have 
v=o, and hence C=o, and 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 315 


r°y 
Oran (1) 
is the measure of a sector of a circle of which v=the meas- 
uring arc. Making R=, we shall have v=the arc of the 
given circle, and equation (1) becomes 
rv 
5s=— 
2 
that is, the measure of a sector of a circle is half the pro- 
duct of the radius into the arc of the sector; and hence for 
the entire circle, the area is equal to half the product of the 
radius into the circumference. 
If we make v= the entire circumference we have 
v=27R 
and substituting this value in equation (1) we have 
rey 
that is the area of a circle is equal to the square of the 
radius multiplied by the ratio between the diameter and the 
circumference. 


(228) To find the area of an Ellipse. 
We have in the case of an ellipse (Art. 64) 
as =yav= Fas a 
hence 
S=5 /(At—x\hee 


Integrating by formula B (Art. 209), and substituting 
} o for m 
A® for @ 
—1 for 
2 for 
4 for p 
we have 


> 


316 INTEGRAL CALCULUS. 


x(A® =a 


SJ (M2 22 \tdx= ee A (A228) tae 


but (Art. 178) 
tks x 
S (A? —x*) %de= =e ae = =sin. oe 
hence 
B ———— SAD 
Sr ee +"Ssin.-15+C 


Estimating from the center where x=o we have S=o, and 
hence C=o. Making then x=A we have | 
ABS 1 oye Bes 
S=—"sin.c11=-= - = 
2 23g 
for one-fourth of the area of the ellipse, since the arc whose 
sine is 1 is equal to one-fourth of the whole circumference; 
and we have for the area of the entire ellipse 
=7AB 
We may also observe that (Art. 63) 
AX A — x? 
is the differential of the area of a circle whose radius is A, 


Pet, ius : 
hence the area of an ellipse is AX the area of the circum- 
scribing circle which is zA®; and is, therefore, equal to 

= 
=~ .7A*=zAB 


If A=B the expression oe 
zA® or zR? 
for the area of a circle. 


(229) Zo find the area of a Segment of a Hyperbola. 
We have inthe case of a hyperbola 
1 ee 
=——“/ ~2—A2 
df ‘A “id 


whence 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 317 


: Be ees 
dS=ydx= Aqev x8—AF 
Integrating by formula € (Art. 209) we have 
B Sa ee BE: sad a Dit St 
Ve Ns log. (v9 +V «2 —A?)+C 
To find the value of C we make x=A where S=o, and we 
have 
AB 
Omar log. A+C 
hence 
AB 
VER pe log. A 
and 
gem) 
A 
which represents a portion of the area between the curve 


and the ordinate lying on one side of the axis. Hence the 
area of the entire segment cut off by the double ordinate is 


(eats) 


B ae AB 
ay AA ert log. ( 


B joe BO AES 
AV x? —A®—AB log. 
but 
The ab Pal ese 
V2? a Me —=y 
hence 
a 
S=xy—AB log. (<+5) =xy—AB log. ( 
for the value of the area. 
- (230) We may also find the area of that part of the sur- 
face lying between the curve and the asymptotes, by using 


the equation of the hyperbola referred to its center and 
asymptotes, which is 


bociBie) 
AB 


xy=m 
but as the asymptotes are not usually at right angles to each 
other, we must introduce into the expression for the differ- 
ential of this area, the s¢we of the angle which they make 


318 INTEGRAL CALCULUS. 


with each other (Art. 58) which we will call v We shall 


then have 


: : max 
@SSsin. v.ydx=sin. Coe 


and 

S=sin.v.m.log.x+C 

If we call the abscissa of the vertex 1, and estimate from 
the corresponding ordinate, we shall have at that point 
m=1, S=o, log. x=o and hence C=o 

And since sin. v may be considered as the modulus of a sys- 
tem of logarithms, we may make 

S=M. log. x=Log. x 
That is, the area between the curve and the asymptote, 
estimated from the ordinate of the vertex, is equal to the 
logarithm of the abscissa, taken in a system whose modulus 
is the sine of the angle made by the asymptotes with each 
other. 


(231) To find the area of a Cycloid. 
We have (Art. 129) 
ad 
= = 5 
V 2ry—y 
hence 
4 
ads = ye =F 
V ary—y? 
Integrating this by formula ¢c (Art. 202) we have 
S=3r. ver. nt oe 2ry—y*® +C 


2 
Estimating the integral from A (Fig. 76) where y=o, we 
have 

S=o and hence C=o 
and taking the integral where 

y=2r=DE 

we have 
are 


S=8r. ver. sin.-127=3 


2 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 319 


that is, the area ADE is equal to three times the semi-circle 


DF’E. Hence:the entire area of the @-_6 D 
cycloid is equal to three times the area 
of the generating circle. p78 


(232) Another method of obtaining 
the area of a cycloid is, to consider that { 
portion of the rectangle ACDE which Fig. 76. 
lies outside the curve. 

If we make GF =27—y=v7 we shall have the differential 
of the area DCAF equal to wdx or 
20S. = =ayV 2ry—y? 2ry—y* 

V ary—y 
Now if we take the equation of a circle with the origin at 
the extremity of the diameter we shall have 

yae= aden yeaah 
which is the differential of the segment of a circle of which 
xis the abscissa. Hence dy 27y—y? 1s the differential of 
a circle of which y is the abscissa, that is of the segment 
F’BE. Hence the two areas ACGF and F’BE have the 
same rate of change or differential for the same value of y ; 
and since they are both equal to zero when y=o, they are 
equal for every other value of y (Art. 173), and, of course, 
when y=27. Hence 


ACD =Dh Hees 


Lm 


aS! =(ar—y)dx=(2r—-y) 


2 


But the rectangle ACDE=7;7 .2r=277?, hence 
= re 


ADE=ACDE—ACD— 

as we found in (Art. 231). | 

(233) Zo find the area bounded by the coordinate axes and 
the logarithmic curve. 


We have had (Art. 137) for the logarithmic curve 


May 
x=Log. y and dk=—— 5 


320 INTEGRAL CALCULUS. 


hence 
@S=ydx=Mady and S=My+C 
If we estimate the area from AD (Fig. 56) where y=1 we 
have 
o=M+C 
whence 
C=—-M 
and 
S=M(y—1) 
If we make y=o we have 
S=—M=area ADD’ 
If y=z=RO we have 
S=M=area ADRO 
So that although the axis of abscissas 1s an asymptote (Art. 
88) to the curve on the negative side, and, therefore, will not 
meet it within a finite distance, yet the area erclosed be- 
tween them is limited and equal to ADRO. 


(234) If we take the curve represented by the equation 
I 
DLae 
to which the axes of coordinates are asymptotes, we shall 
find a case somewhat similar. 
Putting the equation into the form ©, » 


5 
we have 
a 
ik h== tee 
y 
and 
@S=yde=—~ 
== — 
AA ye 

hence 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 321 


If we estimate the area from the line AC where y= o we 
have 


o=o0+C 
hence 


C=onand 4 
cy, 


If we make y=1=FT we have 
S=2=ATDC 

that is, the area ATDC is equal to twice the square AHFT, 
and is, therefore, finite, although the curve FD does not 
meet the axis AC at a finite distance. 

If we take the area between the limits y=1 and y=o, we 
shall have 

S=2—2 

that is, the area FEBT is infinite, although AB is likewise an 
asymptote to the curve. 


(235) Zo find the area described by the radius vector of the 
Spiral of Archimedes, - 


The differential of the area of a polar curve (Art. 82) is 
r°av 
2R 
v being the arc of the fheasuring circle and R its radius. 
The equation of the Spiral of Archimedes (Art. 84) is 


r=—av 
hence 
ar=ado 
or 
ar 
ij eet a, 
a 


Hence, making R=1 we have 
ready re-dr 
s=f Fas ml ae 
If we make r=o we have 
21 


322 INTEGRAL CALCULUS, 


S=o and hence C=o 


: r 
and since as, we have 


fae 
6 
Making v=2z we have for the area described by one revo- 


lution of the radius vector 


mr? 


S 
that is, the area described by one revolution of the radius 
vector is one-third of the area of the circle described with a 
radius equal to the last value of the radius vector. 
If the radius vector make two revolutions we have v=47 
and 


re 

2rr 

Se 
: 3 
where 7 =27 and 

8zr? 
See 

% 


But in making two revolutions, the radius vector describes 
the first part of the area twice. This, therefore, must be 
subtracted, and we have the area enclosed by the curve and 
radius vector after two revolutions equal to 

Sap2 Age? =I gy? 

3 3 3 
and by subtracting the first again we have the zzcreased area 
described during the second revolution equal to 

tar® —trr*® Sarr? 
After m revolutions we have 

mr * 
3 
where 7 =mr, hence 
mrm2r® mrxr® 


‘7 5 ae (7) 


Subtracting from this the area desgribed by the radius vec- 
tor during (#—1) revolutions we have 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 323 


LS a 


+ (m3 —(m—1)°) (2) 
Substituting (#+1) in place of # we have 


: ((m+1)%—m?) (3) 


for the area described by the radius vector during the 
(m-+1)c revolution. Taking the difference between equa- 
tions (2) and (3) we have the additional area described by 
the (m-+1)éé revolution of the radius vector, that is the area 
lying between the mh and (m+1)é& spires thus 


= 


ce — 


118 ood 
((m+1)?—2m? +(m— )3)==2m77 > 


eee 
3 


~ We have found the additional area described by the radius 
vector during the second revolution equal to 27r*, hence the 
additional area described during the (#-+1)é/ revolution is 
equal to # times that described by the radius vector during 
the second revolution. That is, the zzcrease of the additional 
areas described by the radius vector during successive revo- 
lutions, is wazform and equal to twice the area of the circle 
described with a radius equal to the radius vector after one 
revolution. 

If the area ABP (Fig. 34) be required, that is, the add- 
tional area corresponding to the arc BC described after the 


first revolution, we shall have 
27 


pate 
and 
; r 
r ele. 
and the required area will be 
Tv 
uns ree ree 
Sr Aas 


or 


324 INTEGRAL CALCULUS. 
wa(n1) 7" (eae) ee 
rt sy gee TS 

2 
a 
= Shahi 

Developing (z+1)® we have 


tre \ 7 
Steir Us +32 een 


or 


If BCD=+ circumference oes then ee and 
Tv 
ABP=77?(1 +4+ 4) 


(236) Zo find the area of the surface described by the radius 
vector of the Hyperbolic Spiral. 


The equation of the hyperbolic spiral (Art. 86) is 
ab 
rv=ab or Pane 
in which a is the radius of the measuring circle and is the 
unit of the measuring arc. Hence 


s=/[— “dv = {2 @. ab? 
ry yi ae 


which is infinite when v=o, and zero Pie yv—o. If we 
make v=S=AB (Fig. 38), and v=46=As, we shall have 
nO ee C0 eet Dies 


S=a——=—= 
2, 2 2 


(237) Zo find the area described by the radius vector of a 
Logarithmic Spiral. 


We have (Art. 87) for this spiral 
v=Log. 7 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 325 


and 


Mar 
dv=— 


7 
Substituting this value in the formula, and making R=1 and 
M=1, we have 


LV EEREES  V Sos 

= 2 Re eh Vina eieriaess 

If we estimate from the pole where So we have r=o, and 
hence Co, and 


9 


ye 
4 

That is, the area described by the radius vector of the Nape- 

aian logarithmic spiral is equal to one-fourth of the square 

described upon the last value of the radius vector. 


= 


(238) Areas of Surfaces of Revolution. 


We have seen (Art. 66) that the differential of a surface 
of revolution, where the axis of revolution is the axis of 
abscissas, is 

S=/2tyvV/ dx? +dy* 
the radical part being the differential of an arc of the re- 
volving curve. 

To apply. this formula to a particular case we must obtain 
from the equation of the revolving curve, the value of one 
variable and its differential in terms of the other, so that, 
when substituted in the formula we may have the differen- 
tial of the surface in terms of one variable which can then 
be integrated. 


(239) To find the convex surface of a Cone. 


We have (Art. 67) for the differential of the convex sur- 


face of a cone 
2S =27axdxn/ g? +4 


in which « is the length of the axis and @ is the tangent ot 


326 INTEGRAL CALCULUS. 


the angle made by the revolving element of the cone with 
the axis of revolution. Integrating we have 
S=rax*V/q=+14+C 
Estimating from the vertex where S=o we have x=o, and 
hence C=o and 
S=rax*/gr+1 


But from the equation of the generating line we have 


y=ax 
and hence 
2 
9 es 
a=" 3 


Borer 
S=zyx, 5 =ayV x? + 
or (Fig. 35) making x=AB 


S=rCDV AB? +CD? 
that is, the convex surface of a cone is equal to the circum- 
ference of the base multiplied by half the slant height. 
(240) For the convex surface of a cylinder we have 
y=R=radius of the base 
hence 
S=/anyV dx? +dy*? =fenRdx=27Rx 
that is, the convex surface of a cylinder is equal to the cir- 
cumference of its base into its altitude 
(241) In the case of the sphere we have (Art. 68) 
@S=27Rdx 
hence 
S=27Rx«+C 
Estimating from the center where x=o we have S=o and 
hence C=o, and the measure of an indefinite portion of the 
convex surface of a sphere is 
S=e2zRx 
the same as that of the circumscribing cylinder having the 
same altitude. 


MEASUREMENT OF GEOMETRICAL MAGNITUDES, 327 


Making «=R we have 
S=27R? 
for the measure of the convex surface of half the sphere ; 
hence for the entire sphere we have 
S=47R* 
or four great circles. 


(242) Zo find the surface of an Ellipsoid described by an 


ellipse revolving about tts major axts. 


Making 
V dx? + dy? =u 
we have 
anyV dx® +dy? =27ylu 
But we have found (Art. 218) 


Adx Cre ee ea" 3e°x8 
ft agg AT ep tie 6As ete.) 
hence 
| az Ayax eae eet Bit A 
ee eg as hale Hy Nee a ie Auer eo Ata) 
But 
Ew iy ohn 
VA2— x? 
hence 
e* x" e4x4 3e% x6 
A Fae 22 © Sanaa mes ag Aa iach, 
Integrating each term separately we have 
Ayo te Bae Ceae 
peg sae ie icing Wee ey ge Rn ay eee ae 
Taking the integral between the limits 
x=o and «=A 
we have for half the surface of the ellipsoid 
ae gh 3e4 
oeeh a(t a pememaan eee, 7 etc.) 


328 INTEGRAL CALCULUS. 


and multiplying this expression by 2 we have the measure 
of the entire surface of the ellipsoid. 
If we make A=B, then ¢e=o, and we have for the surface 


of the sphere 
S=47k? 
as before. 


(243) Zo find the surface of a Paraboloid of revolution. 
We have (Art. 69) in the case of the paraboloid 
as=anyy/2 TE ay? =F 4 p2yt 
Integrating according to (Art. 165) we have 
S=F (y?-+p*)F+C 
Estimating from the vertex where S=o and y=o we have 


3 == 
= Spree Post 


hence 
amp* 
3 


C=— 
and 


Gedy ear OY ace 


(244) Zo find the area of the surface described by a Cycloid 


revolving about tts base. 
We have (Art. 129) in the case of the cycloid 
V ary—y? 
hence 


V dx + ay? =\/24 


= 
ary ate =O 53 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 329 


and by substitution 
aS=anyV dx? +dy* Sanya / 22 


ere 
or 
A 
S=erV ar {== 
V ary—y? 
But we have found (Art. 205) 
yay 8r 2y 
——_—_ Fy ees een: _e 
Dig cas 3 2r—y 3 2r—~y 


hence 


eS, 2y ,——. 
S=27 ee aS 


If we estimate the surface from the plane passing through 
the middle point of the base we shall have S=o when y=2r, 
hence C=o. Then making y=o we have for half the sur- 
face required 


ee 
S=orV rales —V 2r)=32nr2 


and for the entire surrace sey that quantity. That is, the 
area of the surface described by the revolution of a cycloid 
about its base is equal to twenty-one and one-third times 
that of the generating circle. 


(245) The Cubature of Solids. 


The cubature of a solid ts to find the dimensions of an equiv- 
alent cube or other known volume. 

We have (Art. 71) 

ty® ax 

for the differential of a solid of revolution where y is the 
ordinate and x the abscissa of the bounding line of the 
revolving surface which generates the solid; and the axis of 
abscissas is the axis of revolution. Hence 


V=/xy*dx 


330 INTEGRAL CALCULUS. 


To apply the formula to any particular solid or volume, we 
eliminate one of the variables by means of the equation of 
the bounding curve, thus producing a differential function 
of one variable which may be integrated. 


(246) To find the volume of a Right Cone. 


Making the vertex the origin we have 
y=ax 
for the equation of the bounding line; but a is the tangent 
of the angle made by this line with the axis of the cone, and 
: b 
is equal to 7 where #4 is the radius of the base and % the 


the length of the axis; hence 
b ay 
ya7x and y =Fa Xe 
whence 
b* OF eee 


V=/ry*de= Ty aaa =A SOCES 


Estimating from the origin we have V=o, x=o, and hence 
C=o, and making x= we have for the entire cone 


h 
V=76?— 
3 


that is, the volume of a cone is equal to one-third of the 
product of its base by its altitude, or equal to one-third of 
a cylinder of the same base and altitude. 


(247) To find the volume of a Sphere. 


From the equation of the circle we have 
y2=R2— 2? 
hence 
oe 
Vim fry dx) 7 RE 8) ae ae Bec) EC 


Estimating from the plane passed through the center, where 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 43% 


x=o, we have S=o and C=o, and making «=R we have 
for half the volume of the sphere 

V=&7R3 
and for the entire sphere 

V=4r7R3 
Since the surface of the sphere is equal to 47R® we have 
the volume equal to the surface multiplied by one-third of 
the radius. 


(248) To jind the volume of an Ellipsoid. 


Taking the origin at the extremity of the transverse axis 
we have for the equation of the bounding line or curve 


B2 
y* a (2Ax—x*) 


hence 
Va Sayama f 25(2hx—a? dene g(a? — =)+c 


Estimating from the origin where xo we have she and 
hence C=o; = making x=2A we have 


Vare5(4A2— $A3)=74B2A=87B? . 2A 


that is, the volume of a prolate ellipsoid is equal to two- 
thirds of the volume of a cylinder having the minor axis for 
its diameter, and whose altitude is equal to the major axis. 

If the ellipse is made to revolve about its minor axis we 
should have 

V' =74A?B 
for the volume of an oblate ellipsoid, and hence 
V: V'::7$B?A: c4A2B:: BA 

that is, the volume of a frolate ellipsoid is to that of an 
oblate ellipsoid generated by the same ellipse, as the minor 
axis is to the major axis. 


332 viel INTEGRAL CALCULUS. 
If A=B we have 


as before. 


(249) To find the volume of a Paraboloid. 


In this case we have 
y” =2px 
and 


2 
V=/ry*de= onp/xdx=a27p— =rpx? 


Estimating from the vertex where «=o we have V=o, and 


hence Co, and 
pace 
V=7px* = i 


or, the volume of a paraboloid is equal to half the volume 
of the circumscribing cylinder. 


(250) Zo find the volume of a Soltd described by the revolu- 
tion of a cycloid about its base. 


Since in the case of the cycloid 


we have 


X,=Xo—V ary—y? 
X, = are of a circle of which ~ is radius and y the versed 
sine. 


MEASUREMENT OF GEOMETRICAL MAGNITUDES. 333 


Integrating between the limits y=o and y=2r7 we shall 
have half the volume required; but y=o gives V=o and 
C=o and y—2r7 gives 


Gras 
X,=X,)=7,r 
easy LES 
X_= 2 X= 2 
and 
sr 2 7278 


hence the entire volume is 
V=52"78 
But the volume of the circumscribing cylinder is 
4rr® , amr=8r*73 
Hence the volume of the solid generated by the revolution 
of a cycloid about its base is five-eighths of that of the cir- 
cumscribing cylinder, 


Albee HN Dex, 


GEOMETRICAL FLUXIONS. 


It has been said that the “ reductio ad absurdum” or method 
of exhaustion of the ancient mathematicians contains the 
germ of the differential calculus. This is an error. There 
is nothing in that method that has any affinity to the true 
principle of the calculus. ‘The method of rates, in the sim- 
ple and obvious meaning of the term, is as remote as possi- 
ble from the method of exhaustion. Its demonstrations are 
direct, logical and conclusive. No absurd hypothesis are 
admissible, and therefore no ‘‘ reductio ad absurdum.” There 
is neither exhaustion nor limits, nor any idea to which these 
methods have any sort of affiliation. 

Moreover the true principles of the calculus are so sim- 
ple and so easily applied that if they had occurred to these 
men they could at once have seized and used them without 
the aid of Algebra, and thus have avoided the “tedious and 
operose reductio ad absurdum” altogether. The principles of 
this science have been so exclusively associated with the 
forms of analysis, that it has come to be considered as purely 
analytical in its character. This is indicated by the term 
“calculus” itself, as well as by the terms ‘transcendental 
analysis” and “ calculusof functions.” But the truth is these 

15 = 337 


338 APPENDIX. 


principles are wholly independent of analysis, and may be 
applied as directly to the geometry of Euclid and Archi- 
medes as to that of Descartes. Those propositions that 
require the “ zedious reductio ad absurdum,” or the absurd 
method of the infinitely sided polygon, may be easily and 
directly solved by them without resorting to the abstractions 
of analysis. | | 

These principles are contained in the method of rates, 
which is in fact their development. As applied to geometry 
they are two, viz. : 

first. The rate of increase of any geometrical magnitude, 
while being generated, may be measured by a supposttive increment 
arising from the uniform movement of the generatrix, at the 
existing rate, during a unit of time, in the direction to which tt 
may be then tending; and, therefore, such supposttive increase 
may be taken as a symbol to represent that rate. 

Second. Lf two magnitudes begin to be at the same momeni, 
and the ratio of thetr rates of increase ts constant, the ratio of 
the magnitudes themselves will be constantly the same as that of 
their rates. 

Thus if two persons set out at the same moment and 
place to travel in the same direction, at constant rates, the 
ratio of the distances traveled by each will constantly be the 
same as that of their rates of travel. If one travel twice as 
fast as the other he will always be twice as far from the 
starting point. 

Now to apply these principles to the measurement of geo- 
metrical magnitudes. 


PROPOSITION I. 


(252) Zo find the arca of a Cireéle. 


We will suppose the circle to be generated by the revolu- 


APPENDIX. 339 


tion of radius CA about the center Cat a 
uniform rate. When the radius is in the 
position CA, and revolving toward B, every | 
point in it will zezd to move in a direction 
perpendicular to CA, and hence the point 
A will zexd to describe the line AB tangent Fig. 78. 
to the circle and perpendicular to CA; and if left to its ten- 
dency zwould describe that line at a uniform rate. The line, 
therefore, may be taken as the symbol representing the rate 
of increase or generation of the circumference of the circle. 
But while the point A tends to move in the direction AB, 
every point in the radius CA éends to move in.a direction 
parallel to it, and at rates proportional to their distances 
from the center C. Hence the radius itself, ¢f left to tts ten- 
dency when at CA, would be found at CB, when the point A 
is at B; and the triangle CAB would be generated at a uni- 
form rate during the same time that the line AB is generated. 
The triangle may, therefore, be taken as the symbol of the 
rate at which the area of the circle is generated, and the 
ratio of these symbols is also the ratio of the rates which 
they represent. But the triangle is equal to $CA. AB, that 
is the ratio between the rate of generation of the circumfer- 
ence and that of the area of the circle is half radius; and 
this being constant is the ratio between any part of the cir- 
cumference and the corresponding part of the circle through- 
out their generation, and, of course, when it is completed. 
Hence the area of the circle is equal to half the radius into 
the circumference. 


Proposition II. 


(253) Zo find the area of the convex surface of a Cone. 


Suppose the cone to be generated by the revolution of the 
triangle ADC (Fig. 79) about the axis DC, The hypothe- 


“340 APPENDIX. 


neese AD will generate the convex 
surface, and the point A will gener- 
ate the circumference of the base. 
When the triangle is in the position 
ADC and revolving towards E, the 
point A if left to its zendency at that 
instant would describe the line AE, 
perpendicular to AC, in some unit of 
time, and hence AE may be taken to Hig. 79- 

represent the raze at which the circumference of the base is 
generated. Now every point in the line AD tends to move 
in a direction parallel to AE, and at a rate proportional to its 
distance from the axis DC; hence if left to that tendency 
the line AD would describe the triangle ADE, and be found 
at DE in the same unit of time. Hence ADE (Art. 251) 
may be taken to represent the corresponding rate of 
generation of the convex surface of the cone. But 

ADE AD ; 

ADE=AE.3AD or —\y =", » that is, the ratio between 
the rates of generation of the convex surface of the cone 
and the circumference of its base is constant and equal to 
half its slant height. Hence the ratio between the magni- 
tudes generated will be the same (Art. 251), and their con- 
vex surface divided by the circumference of the base equals 
half the slant height, or, the convex surface equals the cir- 
cumference of the base multiplied by half the slant height. 


Proposition III. 


(254) Zo find the measure of the volume of a Cone. 


The cone being supposed to be generated by the revolu- 
tion of a right angled triangle about one of its sides, which 
becomes the axis of the cone, while the base is generated 
by the other side as its radius; let us suppose the generating 
triangle to have arrived at the position ADC (Fig. 79), the 


APPENDIX. 341 


point A moving towards C. Then every point in the trian- 
gle zends to move in a direction perpendicular to its plane 
and at a rate proportional to its distance from the axis CD; 
so that if AE is taken to represent the line that would be 
described by the point A in a unit of time in consequence 
of that ¢endency, than at the end of the same unit of time 
the line AD would be found at ED, and the triangle ADC, 
at EDC, so that the pyramid DAEC would be the volume 
generated by the triangle, during the same unit of time and 
may therefore (Art. 251) be taken to represent the raz at 
which the cone is generated; while at the same time the 
triangle ACE would be described by the radius AC of the 
base and would, therefore, represent the rate at which the 
base of the cone was generated. But the volume of the 
pyramid DAEC is equal to its base ACE multiplied by one- 
third of its altitude DC. Hence the rate of generation of 
the cone divided by that of its base =a constant quantity. 
Therefore (Art. 251) the cone itself divided by its base is 
equal to the same quantity being one-third of its height — or 
the volume of the cone is equal to its base multiplied by 
one-third of its height. 


ProposiTion IV, 


(255) Zo find the area of the surface of a Sphere. 

Suppose the sphere to be generated by the revolution of 
the semicircle CBD (Fig. 80) about the diameter CD. Then 
the semi-circumference CBD will 
generate the surface of the sphere. 
Now every point in the curve CBD 
tends to move in a direction perpen- A 
dicular to its plane at a rate propor- 
tional to its distance from CD, the 
axis of revolution, and under this 
tendency it would in some unit of time 


342 APPENDIX. 


assume the position of the semi-ellipse CED, generating at 
the same time the convex surface of the ungula CEDB, 
which is, therefore, the symbol (Art. 251) representing the 
rate of generation of the surface of the sphere, while the line 
EB described by the point B, in the middle of the CBD, and 
perpendicular to its plane is the symbol representing the cor- 
responding rate of generation of the circumference of a great 
circle. But the convex surface of ungula is equal toits ex- 
treme height EB, multiplied by the diameter CD, which is, 
therefore, the constant ratio between the rates of generation 
of the surface of the sphere and of the circumference of its 
great circle. The magnitudes themselves are, therefore 
(Art. 251), in the same ratio, and the surface of a sphere is 
equal to its diameter multiplied by the circumference of its 
great circle. 


PROPOSITION V. 


(256) Zo find the measure of the volume of a Sphere. 


The sphere being supposed to be generated by the revo- 
lution of the semicircle CBD about the diameter CD (Fig. 
80), when it is revolving towards the point E, every point in 
it will zeza to move in a direction perpendicular to its plane, 
and at a rate proportional to its distance from CD the axis 
of revolution; and the point B in the middle of the arc 
CBD will tend to describe the line BE perpendicular to the 
plane of CBD, in some unit of time; and would do soif left 
to that tendency. The semicircle CBD, at the end of the 
same unit of time, would be found in the ellipse CED, 
having described or generated the ungula ECBD, which 
may, therefore (Art. 251), be taken asthe symbol! of the rate 
at which the volume of the sphere is generated (Art. 251), 
while EB is the symbol of the rate at which the circumfer- 
ence of its great circle is generated. But the volume of the 


APPENDIX. 343 


ungula is equal to its extreme height EB multiplied by two- 
thirds of the square of the radius, which is, therefore, the’ 
ratio between the rates of generation of the volume of the 
sphere and of the circumference of its great circle. Hence 
the magnitudes themselves are in the same ratio (Art. 251), 
and the volume of the sphere is equal to the circumference 
of its great circle multiplied by two-thirds of the square of 
its radius — or the circumference, multiplied by the diameter 
(equal to the surface), multiplied by one-third of the radius. 


UNIVERSITY OF ILLINOIS-URBANA 
a = 515B854E C001 
| me ELEMENTS OF THE DIFFERENTIAL AND INTEGRA 


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